
 
UNIVERSIDAD COMPLUTENSE DE MADRID 
 
 
FACULTAD DE CIENCIAS FÍSICAS 
 

DISCRETE QUANTUM SYSTEMS: 

COMPLEMENTARITY, PHASES AND ALL THAT 
 
 
MEMORIA PARA OPTAR AL GRADO DE DOCTOR 

PRESENTADA POR 
 

Eulogio Miguel Cuesta Yustas 
 
 
Bajo la dirección del doctor: 
Luís Lorenzo Sánchez Soto 

 
Madrid, 2008
 
 
• ISBN: 978-84-669-3116-8                        ©Eulogio Miguel Cuesta Yustas, 2007                                              


Eulogio Miguel Cuesta Yustas

Discrete quantum systems:

complementarity, phases and

all that

June 11, 2007

Springer


Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Resumen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Sistemas cuánticos discretos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Sistemas de dos niveles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Sistemas de tres niveles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Fase relativa en interacciones átomo-campo . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Átomos de dos niveles interactuando con campos cuánticos . . . 4
1.2.2 Átomos de tres niveles en interacción con campos cuánticos . . . 9

1.3 Rotaciones pequeñas y hamiltonianos efectivos . . . . . . . . . . . . . . . . . . . . 13
1.4 Grado de polarización para campos bimodales . . . . . . . . . . . . . . . . . . . . . 16

2 Discrete quantum systems: a cursory look . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Two-level systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.1 General considerations: Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.2 Systems of qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.3 Phase for a qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.4 Complementarity in the Bloch sphere . . . . . . . . . . . . . . . . . . . . . . 28

2.2 Three-level systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.1 Bloch sphere for a qutrit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.2 Phases for a qutrit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.3 Complementarity for a qutrit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Relative phase in atom-field interactions . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 Two-level atoms interacting with a quantum field . . . . . . . . . . . . . . . . . . 40

3.1.1 Jaynes-Cummings model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.2 Relative-phase for the Jaynes-Cummings model . . . . . . . . . . . . . 42
3.1.3 Dicke model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.4 Relative-phase for the Dicke model . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Three-level atoms interacting with quantum fields . . . . . . . . . . . . . . . . . 58
3.2.1 Three-level atom coupled to a two-mode field . . . . . . . . . . . . . . . 58
3.2.2 The role of atom-field relative phase . . . . . . . . . . . . . . . . . . . . . . . 61


viii Contents

3.2.3 Relative-phase distribution function . . . . . . . . . . . . . . . . . . . . . . . 62

4 Small rotations and effective Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . 69
4.1 Effective Hamiltonians for nonlinear su(2) dynamics . . . . . . . . . . . . . . . 70

4.1.1 Motivation of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.2 Nonlinear small rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1.3 Dispersive limit of the Dicke model . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Effective Hamiltonians for nonlinear su(3) dynamics . . . . . . . . . . . . . . . 73
4.2.1 Three-level systems interacting with quantum fields . . . . . . . . . . 73
4.2.2 The dispersive limit of the Λ configuration . . . . . . . . . . . . . . . . . . 76

4.3 Effective Hamiltonians for nonlinear su(d) dynamics . . . . . . . . . . . . . . . 77
4.3.1 Multilevel systems interacting with a quantum field . . . . . . . . . . 77
4.3.2 Three-photon resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3.3 Multimode fields and resonance conditions . . . . . . . . . . . . . . . . . . 81

5 Relative phase for two-mode fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.1 States with well-defined relative phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.1.1 Differential phase shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.1.2 Classical visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.3 Quantum visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 SU(2) invariance properties of two-mode fields . . . . . . . . . . . . . . . . . . . . 92
5.3 Quantum degree of polarization as a distance . . . . . . . . . . . . . . . . . . . . . 94

5.3.1 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.2 Maximally polarized states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

A Positive operator-valued measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
A.1 Generalized measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
A.2 Example: POVMs for angular variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

B Hilbert-Schmidt and Bures degrees of polarization . . . . . . . . . . . . . . . 115

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119


Introduction

In the emmerging field of quantum information the concept of interference plays a
prominent role. There are also other reasons that justify the importance of this notion:
quantum interference is used to improve the precision of the measuring, to increase
the speed of data transmission in communications, to get a secure distribution of
cryptographic keys, or to the improvement of the resolution in lithography. Even
though some of these applications can become part of our daily reality in a near
future (like the case of cryptography), the principles in which they are based have
been tested in a number of crucial experiments.

From a classical point of view, the natural way of understanding the phenomena
of interference is in terms of phase, which, can be measured without any type of
ambiguity, in experiments that are more than one hundred and fifty years old.

According to the basic principles of quantum mechanics, each observable is repre-
sented by Hermitian operator that acts on the Hilbert space of the system. The search
for an operator represent us the phase of an harmonic oscillator is an old problem.
Although a complete quantum description does not need any operator asociated to
that variable (since the same information is given by a probability distribution func-
tion of this variable), this do not implies that the problem of finding such an operator
does not have importance.

In last years, a lot of works have been dedicated to the analisis of this variable for
the electromagnetic field, however, there are other many fields where the phase is an
essential variable.

On the other hand, in a classical domain, one needs to work with beams of light
that have a well defined phase between modes. (we are refering to the polarization
properties of light).

The main objetive of this Thesis is to present a simple solution to the definition of
a phase operator in some models commonly used in quantum optics and accordingly,
to introduce a consistent description of the polarization properties of light.

The outline of this Thesis can be summarized as follows: in Chapter 1 we describe
the formal framework for discrete systems (two- level and three-level systems) and
the definition of the phase for these systems, which will be used in the next chapters.
In chapter 2 we analize the interaction between two- and three-level atoms and the


x Introduction

electromagnetic field, and introduce a definition of the relative phase operator for
these models. In chapter 3 we use the existence of deformed algebras to show a method
that allow us to obtain effective hamiltonians in a systematic way. In chapter 4 we
analize the phase between two-mode of the fields from a quantum point of view,
and we introduce the concept of visibility, which allows us to study the polarization
properties of ligth including the polarization degree. The last Chapter summarizes
the main obtained results.


1 Resumen

1.1 Sistemas cuánticos discretos

1.1.1 Sistemas de dos niveles

El estado puro más general correspondiente a un sistema de dos niveles o qubit (esṕın
1/2, átomo con dos niveles de enerǵıa, polarización de un fotón, etc...) se representa
como la combinación lineal

|ψ〉 = c1 |1〉 + c2 |2〉 , (1.1)

donde |1〉 y |2〉 son los vectores de una base (llamada, a menudo, computacional) en
el espacio de Hilbert. A menudo, este vector se describe mediante el espinor

|ψ〉 =

(

c2
c1

)

. (1.2)

Puesto que el estado está normalizado a la unidad, y se define salvo fase global, los
coeficientes se pueden representar como

c1 = sin(ϑ/2), c2 = cos(ϑ/2)eiϕ. (1.3)

Cualquier observable en el espacio de Hilbert bidimensional se reduce a una ma-
triz 2×2 compleja que se puede expresar en términos de las matrices de Pauli (y la
identidad)

σ̂1 =

(

0 1
1 0

)

= |1〉〈2| + |2〉〈1|, σ̂2 =

(

0 −i
i 0

)

= i(|1〉〈2| − |2〉〈1|),

(1.4)

σ̂3 =

(

1 0
0 −1

)

= |2〉〈2| − |1〉〈1|, 1̂1 =

(

1 0
0 1

)

= |1〉〈1| + |2〉〈2|.

(1.5)

A menudo resulta cómodo trabajar con los operadores de subida y bajada


2 1 Resumen

σ̂+ =
1

2
(σ̂1 + iσ̂2) = |2〉〈1|, σ̂− =

1

2
(σ̂1 − iσ̂2) = |1〉〈2|, (1.6)

que cumplen las relaciones de conmutación

[σ̂+, σ̂−] = σ̂3, [σ̂3, σ̂±] = ±2σ̂±, (1.7)

distintivas del álgebra su(2).
El paso siguiente es un conjunto de N qubits idénticos. El espacio de Hilbert de

qubits es ahora C
2 ⊗ C

2 ⊗ ... ⊗ C
2 ∼= C

2N , y su base natural está formada por los
vectores {|1〉1 ⊗ ...⊗ |1〉N , ..., |1〉1 ⊗ ...⊗ |2〉j , ..., |2〉1 ⊗ ...⊗ |2〉N}. donde el sub́ınice
j denota el sistema j -ésimo.

Puesto que los qubits son indistiguibles, el estado debeŕıa venir descrito por una
combinación lineal de la forma

|S,m〉 =

√

m!(N −m)!

N !

∑

|2〉1...|2〉m|1〉m+1...|1〉N , (1.8)

donde la suma se efectúa sobre todas las posibles permutaciones. Una vez definidos
los estados colectivos, los operadores colectivos de este sistema de qubits toman la
forma

Ŝ =
1

2

N
∑

j=1

σ̂
j Ŝ± =

N
∑

j=1

σ̂j
± = Ŝ1 ± iŜ2, (1.9)

donde σ̂
j son las matrices de Pauli correspondientes al j -ésimo qubit. Los operadores

Ŝ cumplen también las relaciones de conmutación del álgebra su(2).
En muchas aplicaciones resulta esencial considerar la fase de un qubit. La imagen

de “clásica” de la esfera de Bloch parece identificar ϕ en (1.4) como dicha fase. Sin
embargo, esta aparece como un parámetro asociado al estado y no como una variable
cuántica asociada a un operador autoadjunto. Con esta finalidad, observese que

〈σ̂−〉 = sin(ϑ/2) cos(ϑ/2)eiϕ, (1.10)

por lo que, 〈σ̂−〉 define la exponencial eiϕ (excepto por un factor constante) sobre la
esfera de Bloch. Parece apropiado trabajar la exponencial compleja de la fase Ê del
qubit como la solución de la descomposición polar

σ̂− =
√

σ̂−σ̂+ Ê. (1.11)

Una vez resuelta esta ecuación para Ê se puede obtener un operador de fase φ̂ definido

por Ê = eiφ̂. En la base stándar, las soluciones unitarias de la ecuación anterior son
de la forma

Ê = |1〉〈2| − |2〉〈1|. (1.12)


1.1 Sistemas cuánticos discretos 3

1.1.2 Sistemas de tres niveles

En esta sección, generalizamos los resultados anteriores al caso de un sistema de tres
niveles (qutrit). En la base computacional el estado puro del qutrit se representa como
la combinación

|ψ〉 = c1|1〉 + c2|2〉 + c3|3〉, (1.13)

que puede ser descrita como

|ψ〉 =





c3
c2
c1



 . (1.14)

La normalización de este estado junto con una elección para la fase global nos permite
representar estos coeficentes como

c1 = sin(ξ/2) cos(ϑ/2), c2 = sin(ξ/2) sin(ϑ/2)eiϕ12 , c3 = cos(ξ/2)eiϕ13 , (1.15)

y estas cuatro coordenadas se mueven en los rangos 0 ≤ ϑ, ξ ≤ π, y 0 ≤ ϕ12, ϕ13 ≤
2π. Para trabajar con operadores que actúan sobre el espacio de Hilbert del qutrit,
utilizamos las matrices de Gellmann:

λ̂1 =





0 1 0
1 0 0
0 0 0



 , λ̂2 =





0 −i 0
i 0 0
0 0 0



 , λ̂3 =





1 0 0
0 −1 0
0 0 0



 , λ̂4 =





0 0 1
0 0 0
1 0 0



 ,

(1.16)

λ̂5 =





0 0 −i
0 0 0
i 0 0



 , λ̂6 =





0 0 0
0 0 1
0 1 0



 , λ̂7 =





0 0 0
0 0 −i
0 i 0



 , λ̂8 =





1 0 0
0 1 0
0 0 −2



 .

que constituyen los operadores de traza cero de su(3).
A veces resulta útil trabajar con los operadores

Ŝij = |j〉〈i|, (1.17)

los cuales pueden ser expresados en términos de las matrices de Gellmann como

S21 = |1〉〈2| =
1

2
(λ̂1 + iλ̂2), S12 = |2〉〈1| =

1

2
(λ̂1 − iλ̂2),

S31 = |1〉〈3| =
1

2
(λ̂4 + iλ̂5), S13 = |3〉〈1| =

1

2
(λ̂4 − iλ̂5), (1.18)

S32 = |2〉〈3| =
1

2
(λ̂6 + iλ̂7), S23 = |3〉〈2| =

1

2
(λ̂6 − iλ̂7).

y sus relaciones de conmutación son


4 1 Resumen

[Ŝij , Ŝkl] = δilŜ
kj − δkjŜ

il. (1.19)

Obviamente, los tres operadores “diagonales” Ŝii miden poblaciones, mientras que
los operadores escalera “no-diagonales” Ŝij generan transiciones del nivel i al j. Para
enfatizar esta idea, definimos los operadores de subida y bajada como







Ŝij
+ = Ŝij if j > i,

Ŝji
− = Ŝij if j < i,

(1.20)

en completa analoǵıa con los operadores σ̂± para el qutrit.
De los tres operadores diagonales sólo dos son independientes debido a Ŝ11 +

Ŝ22 + Ŝ33 = 1̂1, por ello trabajaremos con dos operadores independientes de traza cero
construidos como [los cuales constituyen un subálgebra de Cartan maximal]

Ŝ12
z =

1

2
(Ŝ22 − Ŝ11), Ŝ23

z =
1

2
(Ŝ33 − Ŝ22), (1.21)

que miden las inversiones de población entre niveles correspondientes.
Es de esperar que los operadores de inversión sean conjugados a los operadores de

fase del qutrit. De hecho, notemos que (Ŝ12
± , Ŝ12

z ) y (Ŝ23
± , Ŝ23

z ) corresponden a los qubits
1 ↔ 2 y 2 ↔ 3, respectivamente. No obstante, estos qubits no son independientes ya
que la ecuación (1.19) impone un acoplamiento altamente no trivial entre ellos. Por
analoǵıa con la descomposición polar (del qubit), parece apropiado definir

Ŝ12
− =

√

Ŝ12
− Ŝ12

+ Ê12. (1.22)

Aqúı Ê12 = exp(iφ̂12), φ̂12 es el operador hermı́tico que representa a la fase del qubit
1 ↔ 2. La solución unitaria de la ecuación (1.22) viene dada, salvo fase global, por

Ê12 = |1〉〈2| − |2〉〈1| + |3〉〈3|. (1.23)

1.2 Fase relativa en interacciones átomo-campo

1.2.1 Átomos de dos niveles interactuando con campos cuánticos

El modelo de Dicke describe la interacción de una coleción de A átomos idénticos
de dos niveles con un campo cuántico en una cavidad sin pérdidas. Las dimensiones
espaciales del sistema atómico son más pequeñas que la longitud de onda del campo,
por lo que podemos suponer que todos los átomos sienten el mismo campo. El modelo
desprecia la interación dipolo-dipolo entre átomos (o lo que es igual, sus funciones de
onda no se solapan). El hamiltoniano para este modelo (en unidades ~ = 1) es


1.2 Fase relativa en interacciones átomo-campo 5

Ĥ = Ĥ0 + Ĥint, (1.24)

con

Ĥ0 = ω N̂,

(1.25)

Ĥint = ∆ Ŝ3 + g
(

â†Ŝ− + âŜ+

)

.

Aqúı N̂ = â†â + Ŝ3 es el operador número de excitaciones, g es la constante de
acoplamiento (la cual en esta aproximación es la misma para todos los átomos y
puede ser elegida real), y ∆ = ω − ω0 es la desintońıa entre la frecuencia atómica y
la del campo.

Por simplicidad restringiremos nuestra atención al caso de resonancia entre la
frecuencia atómica y la del campo ω0 = ω ≡ ω. Ya que el campo se describe con
el habitual espacio de Fock |n〉f , la base natural para el sistema total es |n,m〉 ≡
|n〉f ⊗ |m〉a. No obstante, es directo probar que

[Ĥ0, Ĥint] = 0, (1.26)

lo que implica que ambas son constantes de movimiento.
El hamiltoniano Ĥ0 (o, equivalentemente, el número de excitaciones N̂) determina

la enerǵıa total proporcionada, por el campo de radiación y el sistema atómico, la
cual es conservada por la interacción. Esto quiere decir que la aparición de m átomos
excitados requiere la aniquilación de m fotones. Por tanto, podemos reescribir la base
total como

|N,m〉 ≡ |N −m〉f ⊗ |m〉a. (1.27)

Con esta terminoloǵıa |N,m− 1〉 significa |N − (m− 1)〉f ⊗ |m− 1〉a.
Siguiendo el esṕıritu de la sección anterior, describiremos la fase relativa átomo-

campo en términos de una descomposición polar de las amplitudes complejas. Con
este fin, introduzcamos los operadores

X̂+ = âŜ+, X̂− = â†Ŝ−, X̂3 = Ŝ3. (1.28)

Estos operadores mantienen la primera relación de conmutación de su(2) en (1.7),
[X̂3, X̂±] = ±X̂±, no obstante la segunda queda modificada de la siguiente manera:

[X̂−, X̂+] = P (X̂3), (1.29)

donde P (X̂3) representa una función polinómica de segundo grado en el operador
X̂3. Esto es un ejemplo t́ıpico de las llamadas deformaciones polinómicas del álgebra
su(2). En particular, el estado |N, 0〉 desempeña el papel de vaćıo cuántico, ya que


6 1 Resumen

X̂−|N, 0〉 = 0. (1.30)

Por lo tanto, podemos construir subespacios invariantes, como en la teoŕıa estándar
del momento angular en la forma

|N,m〉 =
1

N X̂m
+ |N, 0〉, (1.31)

donde N es una constante de normalización. Se puede probar que

X̂D+1
+ |N, 0〉 = 0, (1.32)

confirmando que el número de estados accesibles es D + 1, siendo D el valor mı́nimo
entre el número de total de excitaciones y el número total de átomos.

Como consecuencia, el espacio completo del sistema puede ser dividido en suma
directa H = ⊕∞

N=0HN de subespacios invariantes bajo la acción de los operadores

(X̂+, X̂−, X̂3), además cada uno de ellos tiene un número fijo de excitaciones.
En cada uno de estos subespacios invariantes el operador X̂3 es diagonal, mientras

que X̂+ y X̂− son operadores escalera representados por matrices de dimensión finita.
Esto sugiere introducir una descomposición polar de la forma

X̂− =

√

X̂+X̂− Ê (1.33)

Ahora podemos garantizar que el operador Ê = eiφ̂, representa la exponencial de
la fase relativa, es unitario y conmuta con el número de excitaciones

ÊÊ† = Ê†Ê = 1̂1,

(1.34)

[Ê, N̂ ] = 0.

Por lo tanto, podemos estudiar mejor su restricción a cada subespacio invariante HN ,
la cual denotaremos por Ê(N). Con estas condiciones, la acción del operador Ê(N) en
cada subespacio viene dada por

Ê(N) =
D

∑

m=0

|N,m〉〈N,m+ 1| + ei(D+1)φ(N) |N,D〉〈N, 0|, (1.35)

donde φ(N) es una fase arbitraria.
Para el estado general inicial

|Ψ(0)〉 =
∑

N,M

QN−MAM |N −M,M〉. (1.36)


1.2 Fase relativa en interacciones átomo-campo 7

tenemos

P (N,φ, t) =
1

2π

∣

∣

∣

∣

∣

∣

D
∑

m′,m=0

QN−mAm CN
m′m(t)eim′φ

∣

∣

∣

∣

∣

∣

2

, (1.37)

y entonces llegamos a la distribución de probabilidad:

P (φ, t) =
1

2π

∞
∑

N=0

∣

∣

∣

∣

∣

∣

D
∑

m′,m=0

QN−mAm CN
m′m(t)eim′φ

∣

∣

∣

∣

∣

∣

2

. (1.38)

Este es un resultado básico y compacto que utilizaremos para analizar la evolución
de las propiedades de la fase del modelo de Dicke.

En la figura 1.1 hemos evaluado numéricamente esta distribución P (φ, t) como una
función de φ y el tiempo adimensional gt, para el caso en el que todos los átomos están
inicialmente desexcitados y el campo se encuentra en un estado coherente con varios
valores del número medio de fotones n̄. En todos los casos, cuando τ = 0 tenemos que
CN

m′m(0) = δm′m y

P (φ, t = 0) =
1

2π

∞
∑

N=0

∣

∣

∣

∣

∣

D
∑

m=0

QN−mAm eimφ

∣

∣

∣

∣

∣

2

. (1.39)

Dos comportamientos diferentes son evidentes a partir de estas gráficas. El primero
ocurre en la región de campo débil cuando el número de excitaciones en el sistema es
mucho más pequeño que el número de átomos, N ≪ A. Si, por simplicidad, asumimos
que todos los átomos están desexcitados y que en el estado inicialmente coherente del
campo el número medio de fotones es pequeño, digamos n̄ ∼ 1, entonces podemos
quedarnos solamente con los términos dominantes en la ecuación (1.38), obteniendo

P (φ, t) ≃ 1

2π
{1 + n̄[|C1

00(t)|2 + |C1
01(t)|2 + 2Re(C1

00(t)C
1
01(t)

∗
eiφ)]}e−n̄. (1.40)

Vemos que, debido a la dependencia periódica temporal de los términos CN
m′m(t), esta

distribución es oscilante para todos los tiempos, lo que es corroborado numéricamente
en la figura 1.1.a.

El segundo caso (y quizá más interesante) corresponde a la región de campo fuerte,
el número de átomos A≪ N . Se puede demostrar que los coeficientes CN

m′m(t) pueden
ser aproximados, hasta orden A/

√
n̄, por

CN
m′m(t) ≃ dA

m′m(−ΩN t), (1.41)

donde
ΩN = 2g

√

N −A/2 + 1/2, (1.42)


8 1 Resumen

Fig. 1.1. Representación de la distribución de probabilidad P (φ, t) como función de φ y el
tiempo adimensional gt utilizando contornos grises para el caso de A = 5 átomos inicialmente
desexcitados y el campo en un estado coherente con los valores siguientes del número medio
de fotones: a) n̄ = 1 (campo débil), b) n̄ = 50 (campo fuerte), y n̄ = 5 (campo intermedio).

y dA
m′m son funciones d de Wigner, definidas como los elementos de matriz para las

rotaciones finitas, a partir de los operadores de las representaciones el grupo SU(2)

dA
m′m(ϑ) = dA

mm′(ϑ) = 〈m′|eiϑŜx |m〉, (1.43)

donde m,m′ = 0, 1, . . . , A. El punto ahora es que, en esencia, únicamente un sube-
spacio de dimensión A+ 1 domina la dinámica. Además, poniendo la forma expĺıcita
de estas funciones d, un cálculo simple da


1.2 Fase relativa en interacciones átomo-campo 9

P (φ, t) =
1

2π

∞
∑

N=0

Q2
N

∣

∣

∣

∣

∣

A
∑

m=0

√

A!

(A−m)!m!
[tan(ΩN t/2)]meim(φ−π/2)

∣

∣

∣

∣

∣

2

[cos(ΩN t/2)]2A.

(1.44)
Cuando A≫ 1 y cuando las oscilaciones quedan bien resueltas, se puede realizar

P (φ, t) =

√

A

2π

∑

N

φN (t) exp

[

−A
2

(φ− π/2 + δN )2
]

, (1.45)

donde φN (t) es una fución del tiempo de estructura complicada que lleva información
sobre los colapsos y resurgimientos pero que tiene poco interés para lo objetivos
que nos ocupan , y δN = arg[tan(ΩN t/2)]. Ahora, esta claro que, ya que δN toma
únicamente los valores 0 y π, las distribuciones gaussianas anteriores tienden a tener
dos picos en φ = ±π/2, en acuerdo con las expectativas clásicas. La presencia de
colapsos y resurgimientos es evidente en la figura 1.1.b, lo cual confirma la anterior
evidencia numérica y anaĺıtica. El conocido comportamiento de la cuasi-independencia
del tiempo en la ventana temporal entre colapso y resurgimiento es también clara.
Como podemos ver, la distribución tiende a ser más aleatoria a medida que evoluciona,
aunque los dos picos en ±π/2 se siguen conservando.

En la región intermedia, cuando N ∼ A, el comportamiento es más complejo,
como se muestra en la figura 1.1.c, y no hay aproximaciones anaĺıticas disponibles.

1.2.2 Átomos de tres niveles en interacción con campos cuánticos

Queremos explorar en detalle las propiedades de la fase de un sistema de tres niveles.
Para ser más concretos, consideraremos una configuración Λ configuración, con niveles
de enerǵıa ω1 < ω2 < ω3 y transiciones dipolares permitidas 1 ↔ 3 y 2 ↔ 3, pero no
1 ↔ 2.

El hamiltoniano para este sistema es

Ĥ = Ĥa + Ĥf + V̂ , (1.46)

donde

Ĥa =
∑

i

ωiŜ
ii,

Ĥf = ωaâ
†â+ ωbb̂

†b̂,

V̂ = ga(âŜ13
+ + â†Ŝ13

− ) + gb(b̂Ŝ
23
+ + b̂†Ŝ23

− ). (1.47)

Aqúı Ĥa describe la dinámica del átomo libre y Ĥf representa los modos de frecuencia
ωa y ωb de la cavidad, con operadores de aniquilación â y b̂, respectivamente. Final-
mente, en el término de interacción V̂ , escrito en las aproximaciones dipolar y de onda


10 1 Resumen

rotante, asumimos que la transición 1 ↔ 3 se acopla de manera casi resonante con el
modo a y la transición 2 ↔ 3 se acopla con el modo b, con constantes de acoplamiento
ga y gb que serán considerados números reales.

La base para el sistema total es |i〉a ⊗ |na, nb〉f , donde |na, nb〉f , es la base de Fock
bimodal habitual. No obstante, se puede probar que los dos operadores número de
excitaciones

N̂a = â†â− Ŝ11 + 1̂1, N̂b = b̂†b̂− Ŝ22 + 1̂1, (1.48)

son cantidades conservadas.
Las desintońıas se definen como

∆a = ω31 − ωa, ∆b = ω32 − ωb, (1.49)

con ωij = ωi − ωj . Por lo tanto, el problema puede ser reducido a estudiar la res-

tricción de Ĥint a cada subespacio H(Na,Nb) con valores fijos de los pares de número
de excitaciones (Na, Nb). En cada uno de estos subespacios H(Na,Nb) hay tres vectores
de la base que pueden ser escritos como

|i;na = Na − µi, nb = Nb − νi〉, (1.50)

donde los valores de µi y νi son definidos como

(µ1, µ2, µ3) = (0, 1, 1), (ν1, ν2, ν3) = (1, 0, 1). (1.51)

Notemos que cuando Na = 1 y Nb = 0 o Na = 0 y Nb = 1 algunos estados pueden
tener un número de ocupación de fotones negativo y deben ser eliminados.

Definamos los operadores

X̂13
+ = â Ŝ13

+ , X̂13
z = Ŝ13

z ;

(1.52)

X̂23
+ = b̂ Ŝ23

+ , X̂23
z = Ŝ23

z .

Estos operadores satisfacen la mayoŕıa de las relaciones de conmutación de su(3) con
tal de que una de ella se reescriba como

[X̂13
+ , X̂23

− ] = −Ŷ 12
+ , (1.53)

donde
Ŷ 12

+ = −âb̂†Ŝ12
+ . (1.54)

No obstante, algunas de ellas deben ser modificadas de la forma siguiente

[X̂13
+ , X̂13

− ] = N̂a(1̂1 − 2Ŝ11 − Ŝ22),

[X̂23
+ , X̂23

− ] = N̂b(1̂1 − 2Ŝ22 − Ŝ11), (1.55)

[Ŷ 12
+ , Ŷ 12

− ] = N̂aN̂b(Ŝ
11 − Ŝ22),


1.2 Fase relativa en interacciones átomo-campo 11

lo que corresponde a una deformación polinómica de la álgebra su(3).
Por sencillez, centrémonos primero en la transición permitida 1 ↔ 3 y notemos

que en todo subespacio invariante tridimensional H(Na,Nb) el estado |1;Na, Nb − 1〉
desempeña el papel de vaćıo ya que

X̂13
− |1;Na, Nb − 1〉 = 0. (1.56)

En este subespacio el operador X̂13
z es diagonal y podemos obtener una descom-

posición polar similar a (1.22), a saber

X̂13
− =

√

X̂13
− X̂13

+ Ê13. (1.57)

El operador Ê13 solución de (1.57) puede ser expresado en H(Na,Nb) como

Ê13 = |1;Na, Nb − 1〉〈3;Na − 1, Nb − 1| − |3;Na − 1, Nb − 1〉〈1;Na, Nb − 1|
+ |2;Na − 1, Nb〉〈2;Na − 1, Nb|. (1.58)

De manera evidente, un razomamiento semejante para 2 ↔ 3 da lugar al operador
Ê23 como

Ê23 = |2;Na − 1, Nb〉〈3;Na − 1, Nb − 1| − |3;Na − 1, Nb − 1〉〈2;Na − 1, Nb|
+ |1;Na, Nb − 1〉〈1;Na, Nb − 1|. (1.59)

Centrémonos en la fase relativa entre en modo a y la transición dipolar atómica
1 ↔ 3. La función de distribución de probabilidad de un estado descrito por la matriz
de densidad ˆ̺(t) se define como

P (Na, Nb, φ
13
r , t) = Tr[ˆ̺(t) |φ13

r 〉〈φ13
r |], (1.60)

donde los estados |φ13
r 〉 son los autoestados de (1.58) y el sub́ındice r corre sobre los

tres posibles autovalores, 0, y ±π/2. Esta expresión puede ser interpretada como una
distribución de probabilidad para la fase relativa y los operadores de excitación N̂a y
N̂b. A partir de esta función, podemos deducir la función para la fase relativa como
la distribución marginal

P (φ13
r , t) =

∞
∑

Na,Nb=0

P (Na, Nb, φ
13
r , t). (1.61)

Hemos evaluado esta distribución numéricamente para los tres valores permitidos
de la fase relativa en el caso en él que los átomos se encuentran inicialmente en el estado
fundamental |1〉 y los modos a y b están en un estado coherente con número medio
de fotones n̄a y n̄b, respectivamente. Para simplificar los cálculos, hemos utilizado el
tiempo adimensional


12 1 Resumen

τ =
gat

2π
√
n̄a
, (1.62)

en todas las figuras y hemos supuesto que ga = gb, ya que esta restricción no limita
nuestro estudio.

En la figura 1.2 hemos representado una situación t́ıpica de campo débil, en la
que el número de excitaciones del sistema es pequeño, esto es n̄a ∼ n̄b ∼ 1. El dibujo
muestra un comportamiento casi oscilatorio.

Fig. 1.2. Función de distribución de probabilidad para los seis valores permitidos para la
fase relativa como función del tiempo rescalado τ en el caso de campo débil con n̄a = n̄b = 1.

Quizá es más interesante el caso de la dinámica de campo fuerte, este se da cuando
el número de excitaciones en el sistema es grande y por tanto n̄a o n̄b, o bien los
dos, son grandes. En la figura 1.3 hemos representado las probabilidades de la fase
relativa para número de fotones n̄a = 50 y (a) n̄b = 0.5, y (b) n̄b = 50, con el átomo
inicialmente en el estado fundamental |1〉. Cuando n̄b = 0.5, la distribución P (φ13

0 , t)
(que es la de encontrar un átomo en el nivel |2〉) es casi despreciable, mientras que
P (φ13

± , t) muestra colapsos y resurgimientos. Esto se puede interpretar f́ısicamente de
la siguiente manera: la transición 1 ↔ 3 es intensa debido al proceso estimulado por
el modo a en el que no hay tansferencia de población al nivel |2〉, lo que origina una
oscilación regular del dipolo 1 ↔ 3 con los correspondiente colapsos y resurgimientos
en la fase relativa. La probabilidad de encontrar el átomo en el nivel |1〉, P (φ23

0 , t),
tiende a 1/2 (excepto en los resurgimientos), lo cual confirma que la transición 1 ↔ 3
está casi saturada.

A medida que n̄b crece, la posición de los colapsos y resurgimientos cambia, de
acuerdo a estimaciones stándar. Cuando n̄a = n̄b = 50, P (φ13

0 , t) esta centrado en 1/4,


1.3 Rotaciones pequeñas y hamiltonianos efectivos 13

Fig. 1.3. Función de distribución de probabilidad para los seis valores permitidos para la fase
relativa como función del tiempo rescalado τ en el ĺımite de campo fuerte con: a) n̄a = 50,
n̄b = 0.5 y b) n̄a = 50, n̄b = 50.

mientras que P (φ23
0 , t) esta centrado en 1/2, esto muestra que las poblaciones tien-

den a estar equidistribuidas porque ahora también la transición 2 ↔ 3 está también
saturada.

1.3 Rotaciones pequeñas y hamiltonianos efectivos

Con el fin de hacer la discursión lo más autocontenida posible e introducir las ideas
subyacentes al método, empecemos con el sencillo ejemplo de una part́ıcula de esṕın


14 1 Resumen

s en un campo magnético. El hamiltoniano de este sistema tiene la forma siguiente

Ĥ = ωŜ3 + g(Ŝ+ + Ŝ−), (1.63)

donde g es la constante de acoplamiento y los operadores Ŝ3, Ŝ+, y Ŝ− constituyen
una representación (2s+ 1)-dimensional de la álgebra su(2).

El hamiltoniano (1.63) pertenece a la clase de los llamados hamiltonianos lineales y
admite una solución exacta. Una manera conveniente de obtener la solución es aplicar
la rotación

Û = exp
[

α(Ŝ+ − Ŝ−)
]

, (1.64)

y recordando que eÂB̂e−Â = B̂ + [Â, B̂] + 1
2! [Â, [Â, B̂]] + . . ., el hamiltoniano rotado,

que es unitariamente equivalente al original, se transforma en

Ĥ = ÛĤÛ† = [ω cos(2α)+2g sin(2α)]Ŝ3+
1

2
[2g cos(2α)−ω sin(2α)](Ŝ++Ŝ−). (1.65)

La idea central es elegir ahora el parámetro α que permita cancelar los términos no
diagonales en la ecuación (1.65). Esto puede hacerse tomando

tan(2α) =
2g

ω
, (1.66)

y el hamiltoniano transformado se reduce entonces a

Ĥeff = ω

√

1 +
4g2

ω2
Ŝ3. (1.67)

Ya que este hamiltoniano efectivo es diagonal, el problema dinámico está comple-
tamente resuelto. El hecho crucial es que cuando g ≪ ω podemos aproximar la
ecuación (1.66) por α ≃ g/ω, y entonces la ecuación (1.64) puede ser sustituida
por

Û ≃ exp
[ g

ω
(Ŝ+ − Ŝ−)

]

. (1.68)

Esta rotación pequeña, aproximadamente (esto es, hasta términos de segundo orden
en g/ω) diagonaliza el hamiltoniano (1.63), dando lugar al hamiltoniano efectivo

Ĥeff = ÛĤÛ† ≃
(

ω + 2
g2

ω

)

Ŝ3 (1.69)

lo que de manera evidente coincide con la solución exacta (1.67) despues de expandir
en una serie de g2/ω2. Una aplicación directa de la teoŕıa estándar de perturbaciones
independientes del tiempo a la ecuación (1.63) conduce inmediatamente a los mismos
autovalores y autoestados que da el hamiltoniano (1.69) en el mismo orden de aproxi-


1.3 Rotaciones pequeñas y hamiltonianos efectivos 15

mación. No obstante, este método es completamente operacional y permite evitar
el tedioso trabajo de calcular las correcciones sucesivas como sumas sobre todos los
estados accesibles.

A partir del ejemplo anterior, vayamos un paso más allá tratando el caso más
general de un sistema que admite algunas integrales de movimiento N̂j y cuyo hamil-
toniano de interacción puede ser escrito como

Ĥint = ∆ X̂3 + g(X̂+ + X̂−), (1.70)

donde g es una constante de acoplamiento y ∆ es un parámetro que representa ha-
bitualmente la desintońıa entre frecuencias de subsistemas diferentes (aunque no es
aśı necesariamente). Los operadores X̂± y X̂3 mantienen la primera relación de con-
mutación de su(2), [X̂3, X̂±] = ±X̂±, sin embargo la segunda aparece modificada de
la siguiente manera

[X̂+, X̂−] = P (X̂3), (1.71)

donde P (X̂3) es una función polinómica del operador diagonal X̂3 con coeficientes que
quizá dependan de las integrales de movimiento N̂j . Estas relaciones de conmutación
corresponden a las ya conocidas deformaciones polinómicas de su(2).

Supongamos ahora que por razones f́ısicas (dependiendo del modelo considerado)
se cumple la condición

ε =
g

∆
≪ 1. (1.72)

Entonces, está claro que (1.70) es casi diagonal en la base que diagonaliza X̂3. De he-
cho, una perturbación stándar muestra inmediatamente que las correcciones, a primer
orden, introducidas sobre los autovalores de X̂3 por la parte no diagonal g(X̂+ + X̂−)
son cero y aquellas de segundo orden son proporcionales a ε. Por lo tanto, aplicamos
la siguiente transformación unitaria a (1.70) (la cual, de hecho, es una rotación no
lineal pequeña)

Û = exp[ε(X̂+ − X̂−)]. (1.73)

Después de algunos cálculos conseguimos, hasta orden ε2, el hamiltoniano efectivo

Ĥeff = ∆ X̂3 +
g2

∆
P (X̂3). (1.74)

Esta técnica también proporciona una valiosa herramienta con la que obtener
correcciones a los autoestados de (1.70). En efecto, resulta sencillo probar que estos
autoestados pueden ser aproximados por

|Ψm〉 = Û†|m〉, (1.75)

donde |m〉 son cada uno de los autoestados de X̂3 y Û es la rotación pequeña corres-
pondiente. Debido a que Û y |m〉 no dependen del tiempo, el operador Û puede ser


16 1 Resumen

aplicado a |m〉 como un desarrollo en ε. Por ejemplo, el autoestado |Ψm〉 hasta orden
ε2 toma la forma

|Ψm〉 =

[

1 − ε (X̂+ − X̂−) − ε2

2
(1 + 2X̂+X̂− − X̂2

+ − X̂2
−)

]

|m〉. (1.76)

Esta representación es especialmente ventajosa cuando el espacio de estados del
modelo es un espacio de representación de la álgebra su(2) deformada que se construye
de la manera habitual por la acción del operador creación X̂+; esto es, |m〉 ∼ X̂m

+ |0〉,
donde |0〉 es el vector de pesos mı́nimo que cumple la condición X̂−|0〉 = 0.

Este procedimiento general muestra que todas las transiciones no resonantes
pueden ser adiabáticamente eliminadas pudiendo trabajar con un hamiltoniano efec-
tivo que contiene únicamente transiciones (cuasi) resonantes. El efecto de los términos
no resonantes se reduce a un efecto Stark dinámico (que puede tener una forma bas-
tante complicada). Evidentemente, las transformaciones que generan hamiltonianos
efectivos también cambian las autofunciones, no obstante las correcciones son de orden
ε e independientes del tiempo.

Como ejemplo relevante, apliquemos nuestro método al ya conocido modelo de
Dicke.

Suponiendo ahora que se cumple el ĺımite dispersivo; esto es,

|∆| ≫ Aλ
√
n̄+ 1, (1.77)

donde n̄ es el número medio de fotones del campo, si aplicamos el método con-
siderando los operadores del álgebra deformada (1.29) sobre el hamiltoniano definido
en la ecuación (1.25), éste se transforma en el hamiltoniano efectivo

Ĥeff = ∆ Ŝ3 +
λ2

∆
[Ŝ2

3 − 2(â†â+ 1)Ŝ3 − Ĉ], (1.78)

donde

Ĉ =
A

2

(

A

2
+ 1

)

1̂1, µ =
λ2

∆
. (1.79)

Siendo Ĉ el operador de Casimir para su(2).

1.4 Grado de polarización para campos bimodales

Consideremos una onda plana propagándose en la dirección z, cuyo campo eléctrico
se encuentra en el plano xy. Los parámetros de Stokes valen

S0 = |EH |2 + |EV |2 , S1 = 2 Re(E∗
HEV ) ,

(1.80)

S2 = 2 Im(E∗
HEV ) , S3 = |EH |2 − |EV |2 ,


1.4 Grado de polarización para campos bimodales 17

donde los sub́ındices H y V indican las componentes horizontal y vertical del campo
linealmente polarizado. Resulta sencillo mostrar que S2

1 + S2
2 + S2

3 = 1. Esto quiere
decir que los parámetros de Stokes de cualquier onda plana monocromática están sobre
la esfera de Poincaré. Si las amplitudes del campo fluctúan, el grado de polarización

se define como

Pcl =

√

S2
1 + S2

2 + S2
3

S0
. (1.81)

Desde el punto de vista cuántico, el campo puede ser descrito completamente por los
operadores de amplitud âH y âV . Las relaciones de conmutación de estos operadores
son stándar:

[âj , â
†
k] = δjk , j, k ∈ {H,V } . (1.82)

Los operadores de Stokes se definen como las contrapartes cuánticas de las variables
clásicas (1.80), a saber

Ŝ0 = â†H âH + â†V âV , Ŝ1 = â†H âV + â†V âH ,

(1.83)

Ŝ2 = i(âH â
†
V − â†H âV ) , Ŝ3 = â†H âH − â†V âV ,

y sus valores medios son precisamente los parámetros de Stokes (〈Ŝ0〉, 〈Ŝ〉), donde

Ŝ = (Ŝ1, Ŝ2, Ŝ3). Usando la relación (1.82), se puede observar inmediatamente que
satisfacen las relaciones de conmutación

[Ŝ, Ŝ0] = 0 , [Ŝ1, Ŝ2] = 2iŜ3 , (1.84)

y permutaciones cicĺıcas.
En términos matemáticos, una tranformación de polarización es cualquier tran-

formación generada por los operadores Ŝ. Debeŕıamos notar que el operador Ŝ2 es
el generador infinitesimal de las rotaciones alrededor de la dirección de propagación,
mientras que Ŝ3 es el generador infinitesimal de los cambios en la diferencia de fase
entre los modos. Como indica la ecuación (1.84), estos dos operadores son suficientes
para generar todas los transformaciones SU(2), lo cual en términos experimentales
significa que dichas transformaciones se pueden lograr con una combinación de mod-
uladores de fase y rotadores

La definición stándar de grado de polarización, en función de los operadores de
Stokes es

Psc =

√

〈Ŝ〉2

〈Ŝ0〉
=

√

〈Ŝ1〉2 + 〈Ŝ2〉2 + 〈Ŝ3〉2

〈Ŝ0〉
, (1.85)

donde el sub́ındice sc indica que se trata de una definición semiclásica, ya que imita la
forma de (1.81). Aún cuando éste proporciona una imagen muy intuitiva, tiene varios


18 1 Resumen

defectos que dan lugar a extraños conceptos, tales como el de estados cuánticos con
polarización “oculta”.

Hoy en d́ıa existe consenso en considerar la luz despolarizada como los estados
del campo que continuan invariantes bajo cualquier transformación SU(2). Cualquier
estado que satisfaga esta condición de invariancia también cumplirá la definición
clásica de estado despolarizado, no obstante el razonamiento inverso no es cierto.
Se ha mostrado que el operador de densidad de tales estados despolarizados puede
ser escrito siempre como

σ̂ =
∞

⊕

N=0

λN 1̂1N , (1.86)

donde N denota la variedad de excitación en la cual hay un número exacto N de
fotones en el campo. Todos los coeficientes λN son reales y no negativos y son con-
sistentes con la condición de traza unidad que los operadores de densidad deben
satisfacer

∞
∑

N=0

(N + 1)λN = 1 . (1.87)

Ahora se puede cuantificar el grado de polarización como

P(ˆ̺) ∝ inf
σ̂∈U

D(ˆ̺, σ̂) , (1.88)

donde U denota el conjunto de estados despolarizados de la forma (1.86) y D(ˆ̺, σ̂)
es cualquier medida de distancia (no necesariamente una métrica) entre las matrices
densidad ˆ̺ y σ̂, tal que P(ˆ̺) satisface algunos requerimientos motivados por preceptos
f́ısicos y matemáticos. La constante de proporcionalidad en la ecuación (1.88) debe
ser elegida de tal manera que P esté normalizada a la unidad, esto es, sup ˆ̺ P(ˆ̺) = 1.

Para nuestro problema, imponemos las dos condiciones siguientes:

(C1) P(ˆ̺) = 0 si y sólo si ˆ̺ es despolarizado.
(C2) Las transformaciones unitarias qué conservan la enerǵıa ÛE dejan P(ˆ̺) invari-

ante; esto es, P(ˆ̺) = P(ÛE ˆ̺Û†
E).

Hay autores que exigen queD(ˆ̺, σ̂) sea una métrica. Esto requiere tres propiedades
adicionales:

1. Definida positiva: D(ˆ̺, σ̂) ≥ 0 y D(ˆ̺, σ̂) = 0 si y sólo si ˆ̺ = σ̂.
2. Simetŕıa: D(ˆ̺, σ̂) = D(σ̂, ˆ̺).
3. Desigualdad triangular: D(ˆ̺, τ̂) ≤ D(ˆ̺, σ̂) +D(σ̂, τ̂).

Estas propiedades son bastante razonables, ya que la mayoŕıa de las distancias
utilizadas en mecánica cuántica están basadas en un producto escalar que au-
tomáticamente las cumple.

Para un analisis detallado consideraremos la métrica de Hilbert-Schmidt


1.4 Grado de polarización para campos bimodales 19

DHS(ˆ̺, σ̂) = || ˆ̺− σ̂||2HS = Tr[(ˆ̺− σ̂)2] , (1.89)

la cual ha sido estudiada previamente en contextos de enredo. De acuerdo con la
estrategia general establecida en la definición (1.88), para un estado dado ˆ̺debeŕıamos
encontrar el estado despolarizado que minimiza la distancia

DHS(ˆ̺, σ̂) = Tr(ˆ̺2) + Tr(σ̂2) − 2Tr(ˆ̺σ̂) . (1.90)

Con todo esto en mente, podemos definir el grado de polarización de Hilbert-Schmidt
como

PHS(ˆ̺) = Tr(ˆ̺2) −
∞
∑

N=0

p2
N

N + 1
, (1.91)

el cual viene determinado no sólo por la pureza 0 < Tr(ˆ̺2) ≤ 1, sino también por la
distribución del número de fotones pN .

La distancia Hilbert-Schmidt no es monótonamente decreciente bajo cualquier
mapa completamente positivo que preserve la traza. Esto ha motivado que la co-
munidad cient́ıfica de información cuántica haya identificado la fidelidad como una
aproximación alternativa. Por ello, propondremos como segundo candidato de distan-
cia la fidelidad

F (ˆ̺, σ̂) = [Tr(σ̂1/2 ˆ̺ σ̂1/2)1/2]2 . (1.92)

Un manera habitual de transformar la fidelidad en una métrica es mediante la métrica
de Bures

DB(ˆ̺, σ̂) = 2[1 −
√

F (ˆ̺, σ̂)] . (1.93)

Ya que a mayor fidelidad F (ˆ̺, σ̂), la distancia de Bures DB(ˆ̺, σ̂) es más pequeña,
podemos definir el grado de polarización de Bures como

PB(ˆ̺) = 1 − sup
σ̂∈U

√

F (ˆ̺, σ̂) . (1.94)

Una definición alternativa seŕıa 1 − supσ̂∈U F (ˆ̺, σ̂), la cual surge de manera natural
en el contexto de computación cuántica. Está claro que estas definiciones ordenan los
estados ˆ̺ de la misma manera. Desafortunadamente no hemos encontrado la expresión
general del estado despolarizado σ̂ que da un valor máximo de la fidelidad. Esta tarea
debe ser realizada caso por caso.


2 Discrete quantum systems: a cursory look

This chapter describes quantum systems with d levels (also known as qudits in modern
quantum information). Our purpose is to establish a basic framework to deal with
the different systems we are going to study in this thesis. Fisrt, we consider the two-
level case (qubit), paying special attention to the physical meaning of the operators
associated to these systems, and treating both one simple qubit and a set of identical
noninteracting qubits. We also disscuss possible approaches to describe the phase
of these systems. The second section covers the description of the three-level case
(qutrit), the associated operators and the phases.

2.1 Two-level systems

2.1.1 General considerations: Bloch sphere

The most general pure state of a two-level system (such as a spin 1/2, a two-level
atom, the polarization of a photon, etc) is represented by the linear combination

|ψ〉 = c1 |1〉 + c2 |2〉 , (2.1)

where |1〉 and |2〉 are a basis (usually called, the computational basis) in the Hilbert
space. This vector can be described by the spinor

|ψ〉 =

(

c2
c1

)

. (2.2)

Because |ψ〉 is normalized to unity, the coefficients can be written as

c1 = sin(ϑ/2)eiϕ1 , c2 = cos(ϑ/2)eiϕ2 . (2.3)

Furthermore, since any state is defined up to a global phase, these coefficients can be
recast as

c1 = sin(ϑ/2), c2 = cos(ϑ/2)eiϕ (2.4)

where ϕ = ϕ2 − ϕ1.


22 2 Discrete quantum systems: a cursory look

Any obervable in the bidimensional Hilbert space reduces to a 2×2 complex matrix
that can be always expanded in terms of Pauli matrices (together with the identity)

σ̂1 =

(

0 1
1 0

)

= |1〉〈2| + |2〉〈1|, σ̂2 =

(

0 −i
i 0

)

= i(|1〉〈2| − |2〉〈1|),

(2.5)

σ̂3 =

(

1 0
0 −1

)

= |2〉〈2| − |1〉〈1|, 1̂1 =

(

1 0
0 1

)

= |1〉〈1| + |2〉〈2|,

(2.6)

in the form
ô = o1σ̂1 + o2ô2 + o3σ̂3 + o01̂1, (2.7)

In other words, (σ̂1, σ̂2, σ̂3) and 1̂1 constitute a basis for the operators actuing in this
space, and the coefficients can be obtained as

oj = Tr(ô σ̂j). (2.8)

It is often more convenient to work with ladder operators

σ̂+ =
1

2
(σ̂1 + iσ̂2) = |2〉〈1|, σ̂− =

1

2
(σ̂1 − iσ̂2) = |1〉〈2|, (2.9)

which fullfill the conmutation relations

[σ̂+, σ̂−] = σ̂3, [σ̂3, σ̂±] = ±2σ̂±, (2.10)

characteristic of the su(2) algebra.
The average values of Pauli matrices in the state (2.1) are

〈σ̂1〉 = sinϑ cosϕ,

〈σ̂2〉 = sinϑ sinϕ, (2.11)

〈σ̂3〉 = cosϑ.

In consequence, 〈σ̂〉 can be seen as a vector on the unit sphere, also known as Bloch
sphere. Each point in these sphere corresponds to a pure state. The north pole cor-
responds to the state |2〉 and the south pole to |1〉, while the phase ϕ in these poles
becomes ill defined.

This formalism can be easily extended to mixed states described by a density
matrix ˆ̺. In this case, one can express ˆ̺ in terms of the Bloch vector r = (x1, x2, x3)
as

ˆ̺ =
1

2
(1̂1 + r · σ̂), (2.12)

where r = 2Tr(ˆ̺σ̂), in such way that


2.1 Two-level systems 23

x1 = 2 Re(̺12),

x2 = 2 Im(̺12), (2.13)

x3 = ̺22 − ̺11.

Fig. 2.1. Bloch sphere

It is easy to show that

|r|2 = 2 Tr(ˆ̺2) − 1 ≤ 1, (2.14)

where the equality holds only for pure states. Hence, mixed states can be pictured
as points inside the unit ball. In particular, the maximal mixed state is precisely the
origin

Alternatively, one can employ spherical coordinates and

̺11 =
1

2
(1 − r cosϑ),

̺12 =
1

2
r sinϑeiϕ, (2.15)

̺22 =
1

2
(1 + r cosϑ),

so pure states correspond to r = 1.

2.1.2 Systems of qubits

The next step is the analysis of a collection of N identical qubits. The Hilbert space
is now C

2 ⊗ C
2 ⊗ ...⊗ C

2 ∼= C
2N , and the natural basis is constituted by the vectors


24 2 Discrete quantum systems: a cursory look

{|1〉1⊗ ...⊗|1〉N , ..., |1〉1⊗ ...⊗|2〉N , ..., |2〉1⊗ ...⊗|2〉N}, where the subscript j denote
the j th system.

The state
|2〉1...|2〉m|1〉m+1...|1〉N , (2.16)

has a degeneration
N !

m!(N −m)!
, (2.17)

because permutations of subsystems do not modify the state. Since qubits are indis-
tinguishable, the state should be described by a normalized linear combination of all
the vectors of the form (2.16); i.e.,

|S,m〉 =

√

m!(N −m)!

N !

∑

|2〉1...|2〉m|1〉m+1...|1〉N , (2.18)

where the sum runs over all possible permutations. Once the collective states are
defined, collective operators take the form

Ŝ =
1

2

N
∑

j=1

σ̂
j Ŝ± =

N
∑

j=1

σ̂j
± = Ŝ1 ± iŜ2, (2.19)

where σ̂
j are the Pauli matrices corresponding to the j th qubit. In the collective basis

(2.18)

Ŝ+ |S,m〉 =
√

(m+ 1)(N −m) |S,m+ 1〉,
Ŝ− |S,m〉 =

√

m(N −m+ 1) |S,m− 1〉, (2.20)

Ŝ3 |S,m〉 = (m−N/2) |S,m〉,

where (m−N/2) define the inversion of the system, i.e. the difference between excited
and not excited qubits. The operators Ŝ fulfill the conmutation relations of the su(2)
algebra. Moreover, the state |S,m〉 is an eigenstate of the operator Ŝ2 = Ŝ2

1 + Ŝ2
2 + Ŝ2

3 ,
with eigenvalue

Ŝ2|S,m〉 =
N

2

(

N

2
+ 1

)

|S,m〉, (2.21)

so, the states |S,m〉 form a basis of the (2S+1)-dimensional irreducible representation
of the su(2) algebra with spin S = N/2. The vectors |S,m〉 are usually known as Dicke
states.

2.1.3 Phase for a qubit

In a number of applications the phase of the qubit turns to play an essential role. The
Bloch sphere picture suggest to identify this phase with ϕ in equation (2.4). However,


2.1 Two-level systems 25

this phase ϕ appears as a pure state parameter instead of a variable associated with
a bona fide Hermitian operator. To bypass this drawback, we observe that

〈σ̂−〉 = sin(ϑ/2) cos(ϑ/2)eiϕ, (2.22)

so 〈σ̂−〉 defines (excepts for constant factors) the exponential eiϕ over the Bloch
sphere. It seems thus appropriate to define the complex exponential of the qubit
phase Ê as the solution of the polar decomposition

σ̂− =
√

σ̂−σ̂+ Ê. (2.23)

After this equation is solved, a phase operator φ̂ can be obtained by Ê = eiφ̂. In the
standard basis, the solution of (2.23) reads

Ê = |1〉〈2| + ei2ϕ0 |2〉〈1|, (2.24)

where ϕ0 is a undefined matrix element that appears due to the unitarity require-
ment. Although the main features of this operator are largely independent of ϕ0, its
eigenvectors and eigenvalues depend on ϕ0. For the sake of concreteness, we can make
a definite choice by imposing further conditions. For instance, according to equation
(2.22), the complex conjugation of the wave function should reverse the sign of φ̂.
This leads to ei2ϕ0 = −1 and therefore

Ê = |1〉〈2| − |2〉〈1|, (2.25)

with eigenvalues ±π/2 and eigenvectors

|φ±〉 =
1

2
(|1〉 ± i|2〉). (2.26)

To any function f(φ) we can thus associate the operator

f(φ̂) =
∑

±
|φ±〉 f(φ±) 〈φ±|, (2.27)

with average value

〈f(φ̂)〉 =
∑

±
f(φ±)P (φ±), (2.28)

where P (φ±) is the probability distribution

P (φ±) = Tr(ˆ̺ |φ±〉〈φ±|). (2.29)

Note that this phase can take only two values due to the dimension of the Hilbert
space. This seems anti-intuitive and one may think it preferable to describe the qubit


26 2 Discrete quantum systems: a cursory look

phase by a positive operator valued measure (POVM), taking continuous values in a
2π interval, even though this cannot lead to an operator description (see Appendix
A). This is the approach we examine in the following. Now the shifting property
associated with the qubit phase is

eiφ′σ̂3 ∆̂(φ) e−iφ′σ̂3 = ∆̂(φ+ φ′). (2.30)

The most general POVM fulfilling this property and the statistical conditions of a
POVM, that we can see in Apendix A, is of the form

∆̂γ(φ) =
1

2π
(1̂1 + γeiφσ̂+ + γe−iφσ̂−), (2.31)

here γ ≤ 1 is a real number.
The probability distribution induced by this POVM is

P (φ) =
1

2π
(1 + c eiφ + c∗e−iφ), (2.32)

where c = 〈1| ˆ̺|2〉 γ. Note that

〈σ̂1〉 =
1

γ

∫

2π

dφ cosφ P (φ), 〈σ̂2〉 =
1

γ

∫

2π

dφ sinφ P (φ),

(2.33)

σ̂2
1 = σ̂2

2 = 1̂1/4,

that is, P (φ) contains the complete statistics of σ̂1 and σ̂2: in particular, it contains
the whole statistics of the qubit and not only of its phase.

Supose that qubit is described by two differents POVMs, labeled by γ1 and γ2,
such that γ1 < γ2. If one take the dispersion D as a natural uncertity measure of the
phase

D2
γj

= 1 −
∣

∣

∣

∣

∫

2π

dφ eiφPγj
(φ)

∣

∣

∣

∣

2

= 1 − |cγj
|2, (2.34)

one has Dγ1
≥ Dγ2

: this means that Pγ1
(φ) is broader than Pγ2

(φ) if γ1 < γ2.
Furthermore, one can easily demostrate that

∆̂γ1
(φ) =

1

2π

∫

2π

dφ′
{

1 +
γ1

γ2
[ei(φ−φ′) + e−i(φ−φ′)]

}

∆̂γ2
(φ′), (2.35)

so both POVMs contain the same information; since one of them can be expressed as
a linear combination of the other.

A relevant feature of this approach based on POVMs is that it provides a descrip-
tion where any value for φ is allowed. Nevertheless, this continuous range of variation


2.1 Two-level systems 27

is not completely effective in the sense that the values of P (φ) at every point φ can-
not be independent, and we can find relations between them irrespective of the qubit
state. In other words, all ∆̂γ(φ) cannot be linearly independent because the Hilbert
space is two dimensional and the algebra of operators is four dimensional.

The general P (φ) depends on the complex parameter c . This c can be determined
by the value of P (φ) at two φ points not differing by π. Nevertheless, more manageable
expressions emerge if we use three points instead of two, such as φr = 2πr/3 (with
r = −1, 0, 1). We have

c =
2π

3

∑

r=0,±1

P (φr) e
−iφr , (2.36)

which allows us to express P (φ) as

P (φ) =
1

3

∑

r,s=0,±1

P (φr) e
is(φ−φr), (2.37)

and so the knowledge of the three values P (φr) gives P (φ) at any other point φ.
This effective discreteness allows us to compute the mean values of any function

f(φ) in a way very similar to equation (2.28)

〈f(φ)〉 =
2π

3

∑

r=0,±1

f̃(φr) P (φr), (2.38)

where f̃ is related to f by

∫

2π

dφ eiℓφf̃(φ) =

∫

dφ eiℓφf(φ), ℓ = 0,±1, (2.39a)

∫

2π

dφ eiℓφf̃(φ) = 0, |ℓ| = ±2,±3, . . . ; (2.39b)

Equations (2.39) altogether imply

∫

2π

dφ P (φ) f̃(φ) =

∫

2π

dφ P (φ) f(φ), (2.40)

for any P (φ). Discreteness is then also at the heart of this formalism.
To conclude we just quote that a reasonable choice of POVM could be

∆̂(φ) = |φ〉〈φ|, (2.41)

with

|φ〉 =
1√
2π

(|1〉 + eiφ|2〉). (2.42)


28 2 Discrete quantum systems: a cursory look

While the operator Ê corresponds to a selection of an orthogonal basis from the set
|φ〉, this POVM does not privilege any |φ〉 and all of them play the same role.

The generalization to N qubits is not very difficult by taking into account that
the operators Ŝ± and Ŝ3 constitute a (2S + 1)-dimensional irreducible representation
of the algebra su(2) and, in consequence, the phase take 2S + 1 different values. For
the description in the terms of POVMs, the discreteness emerges too, but now one
need to know the values of P (φ) in 2S + 1 independient points. In other words, the
general form of P (φ) admits only 2S + 1 different frequencies.

2.1.4 Complementarity in the Bloch sphere

One can approach the qubit phase from a different perspective. The idea is that
phase is complementary to amplitude. In physical terms, this means that if one of
these variables is known with precission, the measure of the conjugate variable gives
all the possible outputs with the same probability.

The notion of complementary variables can be appropiately formulated, from a
mathematical point of view, in terms of mutually unbiased basis. Two different or-
thonormal bases A and B are said to be mutually unbiased if a system prepared in
an eigenstate of any element of A (such as |a〉) has a uniform probability distribution
of being found in any element of B (such as |b〉):

|〈b|a〉|2 =
1

d
, ∀|a〉 ∈ A, ∀|b〉 ∈ B, (2.43)

Here d is the dimension of space, and in the limit d→ ∞, is the defining property of
the eigenvectors of position and momentum (2.43).

Given an operator Â, one can assign it the Bloch vector, n = (cosϕA sinϑA,
sinϕA sinϑA, cosϑA) in the form

Â = nA · σ̂ = R̂(ϑA, ϕA) σ̂3 R̂
−1(ϑA, ϕA), (2.44)

with
R̂ (ϑ, ϕ) = exp [i ϑ/2 (cosϕ σ̂1 − sinϕ σ̂2)] , (2.45)

which is the displacement operator over the sphere surface. The condition of comple-
mentarity between Â and a generic operator B̂ [expressed also in the form (2.44)] can
be written as

nA · nB = 0. (2.46)

That is, the subspace spanned by nA in the Bloch sphere is orthogonal to the one
spanned by nB . In this way, we get a one-parameter family of operators

B̂ = nB · σ̂, (2.47)


2.2 Three-level systems 29

where the vector nB fulfills (2.46), which can be written as

cotϑB = − tanϑA cos (ϕB − ϕA) . (2.48)

In particular, if one take amplitude by σ̂3, (so nA = u3) the complementary operators
are determined by all the nB ’s in the equator:

Êϕ0
= cosϕ0 σ̂1 − sinϕ0 σ̂2 =

(

0 eiϕ0

e−iϕ0 0

)

. (2.49)

where ϕ0 represents a reference phase. This, in fact, agrees with the exponential of the
phase operator Ê obtained via a polar decomposition in (2.24) once we set ϕ0 = π/2.

One can see that the transformation from the axis 3 in the sphere to the axis 1,
this is

σ̂1 = F̂ σ̂3F̂
†. (2.50)

is given by the finite-dimensional Fourier transform

F̂ =
1√
2

(

1 1
1 −1

)

. (2.51)

On the contrary, the motions in the 12 are generated by the diagonal operator

V̂ =
1√
2

(

1 0
0 −i

)

, (2.52)

because σ̂2 = V †σ̂1V . In this spirit, one can define a one-parameter family of comple-
mentary operators to the amplitude as

Êϕ0
= eiϕ0σ̂3 σ̂1 e

−iϕ0σ̂3 . (2.53)

which also is agree with (2.24). The principal features of this description are again
independent of the reference phase ϕ0.

2.2 Three-level systems

2.2.1 Bloch sphere for a qutrit

In this section, we generalize the previous results to the case of a three-level system
(qutrit). In the computational basis a pure state of the qutrit is represented by the
linear combination

|ψ〉 = c1|1〉 + c2|2〉 + c3|3〉, (2.54)

that can be described as


30 2 Discrete quantum systems: a cursory look

|ψ〉 =





c3
c2
c1



 . (2.55)

The normalization condition, together with the choice of a global phase, allows us to
express these coefficients as

c1 = sin(ξ/2) cos(ϑ/2), c2 = sin(ξ/2) sin(ϑ/2)eiϕ12 , c3 = cos(ξ/2)eiϕ13 , (2.56)

and these coordinates vary over the ranges 0 ≤ ϑ, ξ ≤ π, and 0 ≤ ϕ12, ϕ13 ≤ 2π.
To deal with operators acting on the three-dimensional qutrit Hilbert space, we use
Gellmann matrices:

λ̂1 =





0 1 0
1 0 0
0 0 0



 , λ̂2 =





0 −i 0
i 0 0
0 0 0



 , λ̂3 =





1 0 0
0 −1 0
0 0 0



 , λ̂4 =





0 0 1
0 0 0
1 0 0



 ,

(2.57)

λ̂5 =





0 0 −i
0 0 0
i 0 0



 , λ̂6 =





0 0 0
0 0 1
0 1 0



 , λ̂7 =





0 0 0
0 0 −i
0 i 0



 , λ̂8 =





1 0 0
0 1 0
0 0 −2



 .

which are traceless operators of su(3) and satisfy

λ̂iλ̂j = 2/3δij 1̂1 + djklλ̂l + ifjklλ̂l. (2.58)

The coefficients fjkl the structure constants of the Lie algebra, given by the com-
mutators of the generators, and are completely antisymmetric in the three indices.
The coefficients djkl are determined by the anti-commutators of the generators and
are completely symmetric. We use latin indices, ranging from 1 to 8, to label these
generators, while greek indices will run over the values 0 to 8.

By supplementing the eight generators with the operator

λ̂0 =

√

2

3
1̂1, (2.59)

we obtain a basis for the space of linear operators in the qutrit Hilbert space, satisfying

Tr(λ̂αλ̂β) = 2δαβ . (2.60)

In consequence, a density operator can be expanded uniquely as

ˆ̺ =
1

3
cαλ̂α, (2.61)

where the (real) expansion coefficients cα are given by


2.2 Three-level systems 31

cα =
3

2
Tr(ˆ̺ λ̂α). (2.62)

Normalization implies that c0 =
√

3/2, so the density operator takes the form

ˆ̺ =
1

3
(1̂1 + c · λ̂). (2.63)

Using equation (2.64) we obtain

ˆ̺2 =
1

9

(

1̂1 +
2

3
c · c

)

1̂1 +
1

3
λ̂ ·

(

2

3
c +

1

3
√

3
c ∗ c

)

. (2.64)

where the product ∗ is given by

(c ∗ d)j = djkl ckdl. (2.65)

For a pure state, ˆ̺2 = ˆ̺, so we must have c · c = 3 and c ∗ c =
√

3c. Defining the
eight-dimensional unit vector n = c/

√
3, we find that any pure state of a qutrit can

be written as

ˆ̺ =
1

3
(1̂1 +

√
3n · λ̂), (2.66)

where n satisfies
n · n = 1, n ∗ n = n. (2.67)

Equation (2.68) implies in this case

nj =

√
3

2
Tr(ˆ̺λ̂j) =

√
3

2
〈ψ|λ̂j |ψ〉. (2.68)

We can obtain expressions for n in the local coordinates (2.56). The pure states lie
on the unit sphere in the eight-dimensional Hilbert space. In fact this vector can be
understood as the Bloch vector for the qutrit, but not all operators on the unit sphere
are pure states. For example, of the unit vectors ui with i = 0, ..., 8, only −u8 satisfies
the star-product condition (2.68) and thus is the unit vector for a pure state. The unit
vectors that do not satisfy the star-product condition do not specify any state, pure
or mixed, for they all give operators that have negative eigenvalues. The star-product
condition (2.68) places three constraints for a pure state, thus reducing the number
of real parameters required to specify a pure state from the seven parameters needed
to specify an arbitrary eight-dimensional unit vector to four parameters, which can
be taken to be the four coordinates of equation (2.56).

It is useful to notice that

|〈ψ|ψ′〉|2 = Tr(ˆ̺ˆ̺′) =
1

3
(1̂1 + 2 n · n′). (2.69)


32 2 Discrete quantum systems: a cursory look

ui Œ {u1, u2, u3} ui Œ {u4, u5; u6, u7}, uj=±u3

Other combinationsui Œ {u4, u5; u6, u7}

u8

ui

uj

ui

u8

ui

uj

ui

Fig. 2.2. Geometry of the 4-dimensional sphere for a qutrit represented as 2-dimensional
sections: Large circles are sections of the ball |n| =

√
3/2. Grey parts are the domain of the

Bloch vector.

Orthonormal pure states have unit vectors n that satisfy n · n′ = 1/2, and are thus
2π/3 apart. The states in an orthonormal basis have unit vectors that lie in a plane
at the vertices of an equilateral triangle. The density operators that are diagonal in
the orthonormal basis are the operators on the triangle or in its interior.

Sometimes it is useful to work with the operators

Ŝij = |j〉〈i|, (2.70)

which can be expressed in terms of the Gellmann matrices as

Ŝ21 = |1〉〈2| =
1

2
(λ̂1 + iλ̂2), Ŝ12 = |2〉〈1| =

1

2
(λ̂1 − iλ̂2),

Ŝ31 = |1〉〈3| =
1

2
(λ̂4 + iλ̂5), Ŝ13 = |3〉〈1| =

1

2
(λ̂4 − iλ̂5), (2.71)

Ŝ32 = |2〉〈3| =
1

2
(λ̂6 + iλ̂7), Ŝ23 = |3〉〈2| =

1

2
(λ̂6 − iλ̂7).

and their conmutation are


2.2 Three-level systems 33

[Ŝij , Ŝkl] = δilŜ
kj − δkjŜ

il. (2.72)

Obviously, the three diagonal operators Ŝii measure populations, while the off-
diagonal ladder operators Ŝij represent transitions from level i to j. To emphasize
this idea, one can alternatively define raising and lowering operators by







Ŝij
+ = Ŝij if j > i,

Ŝji
− = Ŝij if j < i,

(2.73)

in complete analogy with the operators σ̂± for the qubit.
Because of the trivial constraint Ŝ11 + Ŝ22 + Ŝ33 = 1̂1, only two populations can

vary independently. For this reason, it is customary to introduce two independent
traceless operators [which constitute the maximal Abelian or Cartan subalgebra of
su(3)]

Ŝ12
z =

1

2
(Ŝ22 − Ŝ11), Ŝ23

z =
1

2
(Ŝ33 − Ŝ22), (2.74)

that measure inversions between the corresponding levels. In atomic systems, the
selection rules usually rule out one of the transitions and therefore the two indepen-
dent inversions are automatically fixed. For a general qutrit, these inversions can be
arbitrarily chosen.

2.2.2 Phases for a qutrit

We expect the operators Ŝ12
z and Ŝ23

z to be conjugate to the corresponding (indepen-
dent) phases of the qutrit. Note that (Ŝ12

± , Ŝ12
z ) and (Ŝ23

± , Ŝ23
z ) correspond to the two

qubits 1 ↔ 2 and 2 ↔ 3 that exist in the qutrit. However, these two qubits are not
independent, since equation (2.74) imposes highly nontrivial coupling between them.
At the operator level, in view of (2.24) it seems appropriate to define

Ŝ12
− =

√

Ŝ12
− Ŝ12

+ Ê12. (2.75)

Here Ê12 = exp(iφ̂12), φ̂12 being the Hermitian operator representing the phase of
the qubit 1 ↔ 2. The unitary solution of equation (2.75) is given, up to an overall
phase, by

Ê12 = |1〉〈2| + eiϕ12
0 |2〉〈1| − e−iϕ12

0 |3〉〈3|, (2.76)

where the undefined factor eiϕ12
0 appears due to the unitarity requirement of Ê12. For

the sake of concreteness, we can make again a definite choice: the complex conjugation
of the wavefunction reverse the sign of φ̂12, which immediately leads to the condition
eiϕ12

0 = −1. We conclude then that a unitary phase operator that preserves the polar
decomposition of equation (2.76) can be represented as


34 2 Discrete quantum systems: a cursory look

Ê12 = |1〉〈2| − |2〉〈1| + |3〉〈3|. (2.77)

The eigenstates of φ̂12 are

|φ12
0 〉 = |3〉,

(2.78)

|φ12
± 〉 =

1√
2
(|2〉 ± i|1〉),

with eigenvalues of 0 and ±π/2, respectively. This is a remarkable result. It shows that
the eigenvectors |φ12

± 〉 look like the standard ones for a qubit. However, the “spectator”
level |3〉 is an eigenstate of this operator, which introduces drastic changes. In other
words, the phase of the qubit 1 ↔ 2 “feels” the state |3〉.

An analogous reasoning for the transition 2 ↔ 3 gives the corresponding operator
Ê23

Ê23 = |2〉〈3| − |3〉〈2| + |1〉〈1|, (2.79)

with eigenvectors

|φ23
0 〉 = |1〉,

(2.80)

|φ23
± 〉 =

1√
2
(|3〉 ± i|2〉),

and the same spectrum as before.
Finally, for the operator Ê13 one must be careful, because it connects the lowest

to the highest vector. In fact, the polar decomposition in this case gives as a unitary
solution

Ê13 = a|3〉〈2| − b∗|3〉〈1| + b|2〉〈2| − a∗|2〉〈1| + |1〉〈3|, (2.81)

with the condition |a|2 + |b|2 = 1. There are also nonunitary solutions to the polar
decomposition, but they lack the interest to describe a phase observable in our context.

On physical grounds, we argue that the state |2〉 should be a “spectator” for the
transition 1 ↔ 3. Thus we impose Ê13|2〉 ∝ |2〉, which is only possible if a = 0 and
we have that

Ê13 = |1〉〈3| − |3〉〈1| + |2〉〈2|, (2.82)

with eigenvectors

|φ13
0 〉 = |2〉,

(2.83)

|φ23
± 〉 =

1√
2
(|3〉 ± i|1〉).


2.2 Three-level systems 35

We observe that
Ê12Ê23† 6= Ê13, (2.84)

which clearly displays the quantum nature of this phase.
The same reasons for introducing a POVM for the qubit phases can be applied to

the qutrit. If we take into account that

eiφŜ12
z = e−iφ/2|1〉〈1| + eiφ/2|2〉〈2| + |3〉〈3|,

(2.85)

eiφŜ23
z = e−iφ/2|2〉〈2| + eiφ/2|3〉〈3| + |1〉〈1|,

and argue that phase-shift operators must be 2π periodic, we impose that any POVM
∆̂(φ12, φ23) for the qutrit should satisfy

ei2φŜ12
z ∆̂(φ12, φ23) e

−i2φŜ12
z = ∆̂(φ12 + φ, φ23),

(2.86)

ei2φŜ23
z ∆̂(φ12, φ23) e

−i2φŜ23
z = ∆̂(φ12, φ23 + φ).

The general POVM fulfilling these requirements is of the form

∆̂(φ12, φ23) =
1

(2π)2
{1̂1 + [γ12e

i(2φ12−φ23)|2〉〈1| + γ23e
i(2φ23−φ12)|3〉〈2|

+ γ13e
i(φ12+φ23)|3〉〈1|] + c.h.}, (2.87)

where h.c. denotes Hermitian conjugate, γij are real numbers and φij is the relative
phase between states |i〉 and |j〉. If we choose the γij different, say γ12 = 1 and the
other two below the unity, then the expectation value of this POVM could reach the
value zero for the superposition states (|1〉+exp(iϕ)|2〉)/

√
2. However, for superposi-

tions of states |1〉 and |3〉 or |2〉 and |3〉, the expectation values of the POVM would
always be greater than zero. Since there is no physical reason to assign special rele-
vance to one specific superposition, we assume that the POVM must be symmetric
with respect to the states, which leads to

γ ≡ γ12 = γ23 = γ13. (2.88)

Moreover, we make henceforth the choice γ = 1 because only for this choice can the
POVM attain the expectation value zero for some particular state. In contrast with
the result formulated in terms of operators, now there are only two relevant phases in
the qutrit description: the third can be inferred from the other two, as in the classical
description.

The proposed POVM provides qutrit phases where any values of φ12 and φ23 are
allowed. However, note that the probability density induced by this POVM can be
written as


36 2 Discrete quantum systems: a cursory look

P (φ12, φ23) =
1

(2π)2
〈(1 + {c12 exp[i(2φ12 − φ23)] + c23 exp[i(2φ23 − φ12)]

+ c13 exp(i(φ12 + φ23))} + c.c.〉), (2.89)

where cij = 〈i| ˆ̺|j〉 γ and c.c. denotes complex conjugation. Therefore, this continu-
ous range of variation is not effective in the sense that the values of P (φ12, φ23) at
every point (φ12, φ23) are not independent, and we can find relations between them
irrespective of the qutrit state. In other words, the complex parameters cij can be
determined by the values of P (φ12, φ23) at six points. Discreteness is also inevitably
at the heart of the qutrit phase.

2.2.3 Complementarity for a qutrit

We can also analize the qutrit phase from the perspective of complementarity. To
start, we consider the matrices

X̂ =





0 1 0
0 0 1
1 0 0



 , Ẑ =





ω2 0 0
0 ω 0
0 0 1



 , (2.90)

with ω = exp(2πi/3). Note that Ẑ corresponds to σ̂3 with Ẑ|j〉 = ωj |j〉, while X̂
corresponds to σ̂1 with Ẑ|j〉 = |j + 1〉, where the operations must be understood
mod2π/3. These matrices obey the finite-dimensional version of the Weyl commuta-
tion relations:

ẐX̂ = ωX̂Ẑ. (2.91)

and we can generate four classes of disjoint traceless operators, each containing two
commuting operators:

Aχ0 = {Ẑ, Ẑ2}, Bχ1 = {X̂, X̂2},
(2.92)

Cχ2 = {X̂Ẑ, (X̂Ẑ)2}, Dχ3 = {X̂Ẑ2, (X̂Ẑ2)2}.

The eigenvectors of the operators in each one of these classes form mutually unbiased
bases. The two Cartan operators, each associated with the independent inversions

ĥ1 =





1 0 0
0 −1 0
0 0 0



 , ĥ2 =





0 0 0
0 1 0
0 0 −1



 , (2.93)

where ĥ1 and ĥ2 (that correspond to 2Ŝ23
z and 2Ŝ12

z in equation (2.75), respectively)
can be easily expressed as linear combinations of Ẑ and Ẑ2. Note that X̂ and X̂2


2.2 Three-level systems 37

correspond to two different physical situations: in the computational basis X̂ acts as
X̂|j〉 = |j + 1〉, while X̂2|j〉 = |j + 2〉.

Thus, we have two families of commuting phase operators

Êϕ01,ϕ02
= ei(ϕ01ĥ1+ϕ02ĥ2) X̂ e−i(ϕ01ĥ1+ϕ02ĥ2)

(2.94)

Ê2
ϕ01,ϕ02

= ei(ϕ01ĥ1+ϕ02ĥ2) X̂2 e−i(ϕ01ĥ1+ϕ02ĥ2).

The expressions for Ê is

Êϕ01,ϕ02
=





0 ei(ϕ01−2ϕ02) 0
0 0 e−i(2ϕ01−ϕ02)

ei(ϕ01+ϕ02) 0 0



 , (2.95)

where, again, ϕ01 y ϕ02 are reference phases, and coincides in this particular case with
Ê2.

The phase operator (2.77) calculated as a polar decomposition does not coincide
with the operator obtained by complementarity. This is due to the fact the phase
operators from a polar decomposition use the idea of transitions. Indeed, while the
perspective in terms of polar decomposition transitions leads to phase operators that
fulfil the requirements of complementarity only for that between pairs of states in-
volved in each transition, the perspective of complementarity fulfill the requirements
of complementarity in the overall Hilbert state.


3 Relative phase in atom-field interactions

The interaction of an atomic system with a radiation field is a keystone of quantum
optics. Needless to say, it is impossible to obtain exact solutions to this problem and
some approximations must be used; the most common being that the radiation field
is quasimonochromatic and its frequency coincides almost exactly with one of the
transition frequencies of the atoms (supposed identical and with no direct interaction
between them).

The two-level atom is the natural consequence of this hypothesis. The Jaynes-
Cummings model (JCM) consisting of a single-mode radiation field interacting with
the two-level atom is one of the few exactly solvable quantum-mechanical models in
quantum optics. In the framework of this model many nonclassical effects of atom-field
interactions, such as Rabi oscillations, collapse-revival phenomena, or sub-Poissonian
statistics and squeezing of the radiation field have been predicted. Generalizations of
the JCM can be divided into two group. One possible generalization consists in taking
into account the multi-atom system (for instance the so-called Dicke model). Another
generalization, deals with adding other levels . For instance, the three-level atom has
been introduced to study two-photon excitations, coherent population trapping, and
two-photon lasers.

In the semiclassical version of these models, correlations are safely ignored and
the field is interpreted to be a purely classical electric field (Cives-Esclop et al. 1999;
Kumar and Mehta 1970; Stroud and Jaynes 1970). Such an approximation has proven
to be very successful and has the virtue of reducing the problem to the exclusive
knowledge of the atomic dynamics, which is studied in terms of the Bloch vector.

For some phenomena, such as spontaneous emission by a fully excited atomic
system, the quantization of the field is required. Then, one must take care also of the
evolution of the field amplitudes, but the atomic dynamics is still explained in terms
of inversion and dipole quadratures.

The natural way of understanding the resonant behavior of these models is in terms
of the relative phase between the field and the atomic dipole (Ashcroft and Mermin
1996). While the quantum quadratures are well the operator for this relative phase
has resisted a quantum description. At this respect, we think that, in spite of its ma-
turity and success, these models are apparently incomplete since it lacks a satisfactory


40 3 Relative phase in atom-field interactions

description in terms of this relative phase, indispensable to compare with the classical
world.

When focusing on the relative phase between two subsystems, we have two ways
of proceeding: One is to start from previous descriptions of the field and dipole phases
and manage them until we get the probability distribution for their difference. Al-
ternatively, we can focus directly on the relative phase variable, trying to define the
corresponding operator without any previous assumption about either the field or
dipole phase descriptions. We analyze this problem in great detail in this chapter.

In section 2.1 we analyze the interaction of two-level atoms with a quantum single-
mode field and the role played by the relative phase in these systems. First we study
the interaction of a two-level atom with quantum field and in the following we extend
the study to a collection of identical two-level atoms. In section 2.2 we analize the
interaction of a three-level atom with a quantum two-mode field considering, once
again, the role played by the relative phase.

3.1 Two-level atoms interacting with a quantum field

3.1.1 Jaynes-Cummings model

The Jaynes-Cummings model describes the interaction of a two-level atom, with res-
onance frecuency ω0, with a single-mode radiation field of frecuency ω in a cavity. In
the electric-dipole and rotating wave approximation the Hamiltonian of this model
can be expressed in the form (in units ~ = 1)

Ĥ = ω â†â+ ω0 σ̂3 + g(â†σ̂− + âσ̂+), (3.1)

where g is the atom-field coupling constant, and σ̂+, σ̂− and σ̂3 are the Pauli operators
acting on atomic states. The photon creation and annihilation operators â† and â,
with commutator [â, â†] = 1̂1 , act on photon number states |n〉f , eigenstates of the
photon number operator â†â,

â† |n〉f =
√
n+ 1|n+ 1〉f , â |n〉f =

√
n|n〉f . (3.2)

The Hamiltonian (3.1) can be rewritten as

Ĥ = Ĥ0 + Ĥint, (3.3)

with

Ĥ0 = ω N̂,

(3.4)

Ĥint = ∆ σ̂3 + g(â†σ̂− + âσ̂+),


3.1 Two-level atoms interacting with a quantum field 41

where N̂ = â†â+ σ̂3 is the total excitation number, and ∆ = ω0 − ω is the detuning
between the field and atom frecuencies.

It is straightforward to check that

[N̂ , Ĥ] = 0, (3.5)

so the problem can be can be diagonalized in the subspaces with fixed N̂ .
The allowed values for N̂ , when atoms are initially in the ground state, are n−1/2

with n = 0, 1, ...,∞. The corresponding subspaces Hn are spanned by the common
eigenvectors of σ̂3 and â†â:
{|−, n〉 ≡ |−〉 ⊗ |n〉f , |+, n− 1〉 ≡ |+〉 ⊗ |n− 1〉f} for n > 0 and |−, 0〉 ≡ |+〉 ⊗ |n〉f for
n = 0, denoting by |−〉 and |+〉 the ground and excited levels of the atom.

For simplicity we restrict henceforth our attention to case of exact resonance be-
tween the atomic and the field frequency ω0 = ω ≡ ω. The eigenstates in each one of
the subspaces Hn are

|Ψ (0)
0 〉 = |−, 0〉 n = 0,

(3.6)

|Ψ (n)
± 〉 =

1√
2
(|−, n〉 ± |+, n− 1〉) n > 0.

and the respective eigenvalues

h
(0)
0 = −ω/2,

(3.7)

h
(n)
± = [(n− ω/2)ω ± g

√
n].

A first interesting and simple example is the case where the initial state of the
system is the product of the atom in its ground state and the field in a number state
|Ψ(0)〉 = |−, n〉 with n > 0. At later times t, the state is given by

|Ψ(t)〉 = cos
(

g
√
n t

)

|−, n〉 − i sen
(

g
√
n t

)

|+, n− 1〉, (3.8)

and the population inversion Ŵ = |+〉〈+| − |−〉〈−| = 2 σ̂3 is then

〈Ŵ 〉 = − cos(2 g
√
n t). (3.9)

Other interesting example is when the initial state of the field is a coherent state
|α〉 and the atom is in its ground state. At later times t population inversion is given
by

〈Ŵ 〉 = −
∞
∑

n=0

Qn cos(2 g
√
nt) = Re

[ ∞
∑

n=0

Qne
−i2 g

√
nt

]

. (3.10)


42 3 Relative phase in atom-field interactions

where Qn being the Poissonian weighting factor of the coherent state.
Oscillations which occur with photon number states, do not persist indefinitely

when the field is initially prepared in a coherent state. Instead, evolution displays
collapses and revivals of these oscillations.

3.1.2 Relative-phase for the Jaynes-Cummings model

In the spirit of our previous chapter, we describe the atom-field relative phase in terms
of a polar decomposition of the complex amplitudes. To this end, let us introduce the
operators

X̂+ = â σ̂+, X̂− = â† σ̂−,

(3.11)

X̂3 = σ̂3.

These operators maintain the first commutation relation of su(2) in (1.10), [X̂3, X̂±] =
±X̂±, but the second one is modified in the following way:

[X̂−, X̂+] = P (X̂3), (3.12)

where P (X̂3) is a second-order polynomial of the operator X̂3. This is a typical ex-
ample of a polynomial deformation of the algebra su(2). Without embarking us in
mathematical subtleties, the essential point for our purposes here is that one can
develop a theory in a very close analogy with the standard su(2) algebra, that we
analyzed the in previous chapter. In particular, it is clear that the state |−, n〉 plays
the role of a ground state, since

X̂−|−, n〉 = 0. (3.13)

Then, we can build invariant subspaces, as in the usual theory of the angular momen-
tum, since

|+, n− 1〉 ∝ X̂+|−, n〉. (3.14)

One can check that
X̂2

+|−, n〉 = 0, (3.15)

confirming that the number of accessible states is 2.
In consequence, the whole space of the system can be split as the direct sum H =

⊕∞
n=0Hn of subspaces invariant under the action of the operators (X̂+, X̂−, X̂3), and

each one of them having a fixed number of excitations. These independent subspaces
do not overlap in the evolution, in such a way that if the initial state belongs to one
of them, it will remain in that subspace for all the evolution.


3.1 Two-level atoms interacting with a quantum field 43

In each one of these invariant subspaces the operator X̂3 is diagonal, while X̂+

and X̂− are ladder operators represented by finite-dimensional matrices (in analogy
with σ̂±). This suggests to introduce a polar decomposition in the form

X̂− =

√

X̂+X̂− Ê (3.16)

where the radial operator

√

X̂+X̂− is diagonal in the basis {|−, n〉, |+, n− 1〉}, and

[X̂3, Ê] = Ê. (3.17)

Once the polar decomposition (3.16) has been solved in each one of these sub-
spaces, obtaining the operators Ê(n), the solution for the whole space is

Ê =

∞
∑

n=0

Ê(n), (3.18)

from which a Hermitian relative-phase operator φ̂ can be defined as Ê = eiφ̂. The
solutions are

Ê(0) = |−, 0〉〈−, 0|,
(3.19)

Ê(n) = |−, n〉〈+, n− 1| − |+, n− 1〉〈−, n|,

with eigenvectors

|φ(0)
0 〉 = |−, 0〉, for n = 0,

(3.20)

|φ(n)
± 〉 =

1

2
(|−, n〉 ± i|+, n− 1〉), for n > 0,

and eigenvalues

φ
(0)
0 = 0,

(3.21)

φ
(n)
± = ±π

2
. for n > 0,

We have that Ê(n)† = Ê(n) for n > 0, Ê(0)† = Ê(0) for n = 0, and therefore
cos φ̂ = 0 outside H0, and sin φ̂ = 0 for H0. Another striking feature of this result
is that the relative phase can take only three values. This may be surprising since
any value for the field phase seems allowed. The reasons for these behaviors are


44 3 Relative phase in atom-field interactions

the same as those discussed for the qubit phase. Because the operator splits into
components acting on two-dimensional subspaces Hn (one dimensional for H0), the
previous features can be ascribed to the particular dimension of the atomic space.
This is supported by the fact that this operator behaves properly when considering
classical limits for either the atom or the field.

Another relevant point is that Ê cannot be written as a product of phase expo-
nentials for each system. This relative phase is not the difference of absolute phases,
and it does not have the usual mathematical properties of a difference. This is not
exclusive of this formalism, and it also arises in other relative-phase approaches
(Torgerson and Mandel 1996).

For any state, the information one can reap using a measurement of some observ-
able is given by the statistical distribution of the measurement outcomes. For the
relative phase, it seems natural to define the probability distribution function of a
state, described by the density matrix ˆ̺, as

P (0, φ
(0)
0 , t) = 〈φ(0)

0 | ˆ̺(t)|φ(0)
0 〉

(3.22)

P (n, φ
(n)
± , t) = 〈φ(n)

± | ˆ̺(t)|φ(n)
± 〉.

From it, we can derive the distribution for the relative phase as

P (φ, t) =
∑

n=0

P (n, φ, t). (3.23)

For the previous example, where the initial state of the system is the product of
the atom in its ground state and the field in a number state |Ψ(0)〉 = |−, 0〉 with
n > 0, the probability of finding the sistem with the relative phases 0, −π/2 or π/2,
at later t times is

Pn=0(0, t) = 0,

(3.24)

Pn(−π/2, t) = cos2
(

g
√
nt− π

4

)

, Pn(π/2, t) = sin2
(

g
√
nt− π

4

)

.

This gives the mean value
〈Ê〉 = −i sin(2 g

√
nt), (3.25)

and so 〈cos φ̂〉 = 0.
The close relation (simply a time translation) between the evolution of the relative

phase (3.25) and the population inversion (3.9) of the previous example allows us to
expect similar collapses and revivals for the relative phase as those experienced by
the population inversion for the coherent state. When the initial state is |−〉|ψ〉, with
|ψ〉 an arbitrary field state, we have that the mean value of the exponential is


3.1 Two-level atoms interacting with a quantum field 45

〈Ê〉 = Q0 − i

∞
∑

n=1

Qn sin(2 g
√
nt), (3.26)

where Qn is the field photon-number distribution. Equivalently we have

〈sin φ̂〉 =

∞
∑

n=0

Qn sin(2 g
√
nt) = Im

[ ∞
∑

n=0

Qne
−i2 g

√
nt

]

, (3.27)

and 〈cos φ̂〉 = Q0. This relative-phase evolution can be compared with the population
inversion (3.10)

Next, we outline a plausible physical interpretation of the similarity between rela-
tive phase and population inversion. The interaction Hamiltonian (3.4) is proportional
to cosφ. In classical terms, the dipole energy is maximum or minimum either when
sinφ = 0 or when the field quadratures or the atomic dipole components vanish. In
the quantum case, for the initial state |−, n〉, the population inversion has maximum

or minimum values precisely when 〈cos φ̂〉 = 0. This relation holds very approximately
when the initial state is of the field is a coherent state.

If the atom is initially in its ground state, the mean value of the atomic dipole
operator d̂ vanishes and therefore so does the interaction Hamiltonian. Since in the
resonant case the interaction Hamiltonian is a constant of the motion, we would
expect cosφ = 0 and senφ = ±1 at all later times. The relative phase is effectively
uniform at t = 0 due to the randomness of the dipole phase. Due to the quantum
fluctuations, the condition senφ = ±1 cannot be established instantaneously, whereas
this is possible classically. Nevertheless, the trend to satisfy this phase relation can be
observed in the initial stages of the evolution, before the quantum evolution displays
its complexity.

On the other hand, we have that 〈cos φ̂〉 = Q0, and so arg〈Ê〉 ≈ ±π/2 will occur

only provided that 〈sin φ̂〉 ≫ Q0.
Therefore expressions (1.27) and (1.28) show that the previously discussed rela-

tionship between relative phase and population inversion extends to the quantum
case.

3.1.3 Dicke model

The Dicke model describes the interaction of a collection of A identical two-level
atoms with a quantum single-mode field in a lossless cavity. The spatial dimensions
of the atomic system are smaller than the wavelength of the field, so all the atoms
feel the same field. The model neglects the dipole-dipole interaction between atoms
(i.e., their wavefunctions do not overlap in the evolution).

The Hamiltonian for this model reads as

Ĥ = Ĥ0 + Ĥint, (3.28)


46 3 Relative phase in atom-field interactions

with

Ĥ0 = ω N̂,

(3.29)

Ĥint = ∆ Ŝ3 + g
(

â†Ŝ− + âŜ+

)

.

Here
N̂ = â†â+ Ŝ3 (3.30)

is the excitation number operator.
For simplicity we restrict again to the case of exact resonance between the atomic

and the field frequencies ω0 = ω ≡ ω. Since the field mode is described in the usual
Fock space |n〉f , the natural bare basis for the total system is |n,m〉 ≡ |n〉f ⊗ |m〉a.
However, it is straightforward to check that

[Ĥ0, Ĥint] = 0, (3.31)

so both are constants of motion. The Hamiltonian Ĥ0 (or, equivalently, the excitation
number N̂) determines the total energy stored by the radiation field and the atomic
system, which is conserved by the interaction. This means that the appearance of m
excited atoms requires the annihilation of m photons. This allows us to factor out
exp(−iĤ0t) from the evolution operator and drop it altogether. Hence, we can relabel
the total basis as

|N,m〉 ≡ |N −m〉f ⊗ |m〉a. (3.32)

With this terminology |N,m− 1〉 means |N − (m− 1)〉f ⊗ |m− 1〉a.
In such a basis, the interaction Hamiltonian, for a fixed value of N , is represented

by the tridiagonal matrix

Ĥ
(N)
int = g











0 h0 0 . . . .
h0 0 h1 0 . . .
0 h1 0 h2 . . .
...

...
...

. . .
...











, (3.33)

with
hm =

√

(m+ 1)(N −m)(A−m). (3.34)

The dimension of this matrix depends on whether A > N or A < N , which are
situations essentially different and must be handled separately.

Let us assume that A > N and initially all the atoms are unexcited. Then, m = 0
and the conservation of the number of excitations implies that only the states (3.32)
with 0 ≤ m ≤ N take part in the dynamics. Thus, the dimension of the subspace is
N + 1.


3.1 Two-level atoms interacting with a quantum field 47

On the contrary, when A < N the number of initial photons is greater than the
number of atoms and only the states (3.32) with 0 ≤ m ≤ A are involved in the
evolution. The dimension is now A+ 1

It is easy to check that, due to the properties of the tridiagonal matrices, the
eigenvalues are distributed symmetrically with respect to zero, with one eigenvalue
equal to zero if there are an odd number of them (Tanaś et al. 1991).

To find the state evolution we shall need the following matrix elements of the
evolution operator

CN
m′m(t) = 〈N,m′| exp[−iĤ(N)

int t]|N,m〉, (3.35)

which can be written as

CN
m′m(t) =

D
∑

J=0

UmJU
†
m′J exp[−iε(N)

J t], (3.36)

where Û is the unitary matrix that diagonalizes the Hamiltonian and ε
(N)
I are the

corresponding eigenvalues. In what follows we shall use the convention of denoting

the dimension of the Hamiltonian matrix Ĥ
(N)
int by D + 1, that is,

D = min(N,A). (3.37)

Now, let us assume that the initial field is taken to be in a coherent state |α〉f
and that the atomic state is initially prepared in an atomic coherent state |ζ〉a
(Arecchi et al. 1972; Perelomov 1986); i.e.,

|Ψ(0)〉 = |α〉f ⊗ |ζ〉a, (3.38)

where
|α〉f =

∑

n

Qn |n〉f , (3.39)

Qn being the Poissonian weighting factor of the coherent state (with zero phase) with
mean number of photons n̄

Qn =

√

e−n̄
n̄n

n!
; (3.40)

and

|ζ〉a =
1

(1 + |ζ|2)A/2

A
∑

m=0

√

A!

m!(A−m)!
ζm|m〉a ≡

A
∑

m=0

Am |m〉a, (3.41)

where the parameter ζ is normally rewritten in terms of the spherical angles as


48 3 Relative phase in atom-field interactions

ζ = − tan(ϑ/2)e−iϕ. (3.42)

In other words, the initial state can be rewritten, taking into account (3.32), as

|Ψ(0)〉 =
∑

N,m

QN−mAm|N,m〉. (3.43)

With this initial condition the resulting state can be recast as

|Ψ(t)〉 = exp(−iĤintt)|Ψ(0)〉 =

∞
∑

N=0

D
∑

m′,m=0

QN−mAm CN
m′m(t) |N,m′〉. (3.44)

If the initial state is not of the same form, but it has a decomposition with different
amplitudes Am or Qn, equation (3.44) is still valid when the appropriate coefficients
are taken.

3.1.4 Relative-phase for the Dicke model

In the spirit of the previous sections, we shall describe the relative phase between
a system of atoms and the field in terms of a polar decomposition of the complex
amplitudes. To this end, we introduce the operators

X̂+ = âŜ+, X̂− = â†Ŝ−,

(3.45)

X̂3 = Ŝ3.

These operators maintain the first commutation relation of su(2) in (1.10), [X̂3, X̂±] =
±X̂±, but the second one is modified in the following way:

[X̂−, X̂+] = P (X̂3), (3.46)

where P (X̂3) represents a second-order polynomial function of the operator X̂3. This
is again a polynomial deformation of the algebra su(2). Now the state |N, 0〉 plays the
role of a vacuum state, since

X̂−|N, 0〉 = 0. (3.47)

Then, we can construct invariant subspaces, as in the usual theory of angular mo-
mentum, by

|N,m〉 =
1

N X̂m
+ |N, 0〉, (3.48)

where N is a normalization constant. One can check that

X̂D+1
+ |N, 0〉 = 0, (3.49)


3.1 Two-level atoms interacting with a quantum field 49

confirming that the number of accessible states is D + 1.
In consequence, the whole space of the system splits again as the direct sum

H = ⊕∞
N=0HN of subspaces invariant under the action of the operators (X̂+, X̂−, X̂3),

and each one of them having a fixed number of excitations.
In each one of these invariant subspaces the operator X̂3 is diagonal, while X̂+ and

X̂− are ladder operators represented by finite-dimensional matrices. This suggests to
introduce a polar decomposition in the form (3.16)

X̂− =

√

X̂+X̂− Ê (3.50)

We can guarantee now that the operator Ê = eiφ̂, representing the exponential of
the relative phase, is unitary and commutes with the excitation number

ÊÊ† = Ê†Ê = 1̂1,

(3.51)

[Ê, N̂ ] = 0.

Thus, we may rather study its restriction to each invariant subspace HN , we shall
denote by Ê(N). It is easy to check that the action of the operator Ê(N) in each
subspace is given by

Ê(N)|N,m〉 = |N,m− 1〉,
(3.52)

Ê(N)†|N,m〉 = |N,m+ 1〉.

Obviously, the action of Ê(N) and Ê(N)† becomes undefined on the marginal states
|N,D〉 and |N, 0〉. Therefore, it is necessary to add some conventions for closing the
actions of these operators on the subspace HN . By analogy once again with the usual
su(2) algebra, we shall use standard cyclic conditions and impose (up to global phase
factors)

Ê(N)|N, 0〉 = |N,D〉,
(3.53)

Ê(N)†|N,D〉 = |N, 0〉.

With these conditions, the operator Ê(N) can be expressed as

Ê(N) =

D
∑

m=0

|N,m〉〈N,m+ 1|

+ ei(D+1)φ(N) |N,D〉〈N, 0|, (3.54)


50 3 Relative phase in atom-field interactions

φ(N) being an arbitrary phase. Note that the crucial extra term in this equation,
which establishes the unitarity of Ê(N), is precisely based on the finite number of
states. Therefore, in each invariant subspace HN there are D + 1 orthonormal states
satisfying

Ê(N)|φ(N)
r 〉 = eiφ(N)

r |φ(N)
r 〉, (3.55)

with r = 0, . . . ,D. These states can be expressed as

|φ(N)
r 〉 =

1√
D + 1

D
∑

m=0

eimφ(N)
r |N,m〉, (3.56)

and, by taking the same 2π window in each subspace, we have

φ(N)
r = φ0 +

2πr

D + 1
, (3.57)

and φ0 is a fiducial or reference phase shift that can be arbitrarily chosen. The ex-
pression for Ê on the whole space is

Ê =

∞
∑

N=0

Ê(N) =

∞
∑

N=0

D
∑

r=0

|φ(N)
r 〉 eiφ(N)

r 〈φ(N)
r |. (3.58)

In the limit D ≫ 1, the spectrum becomes dense, as it might be expected. But, on the
opposite limit, one may be surprised to find that the state |0, 0〉 is a relative-phase
eigenstate (with arbitrary eigenvalue φ0). While this may provide a convincing argu-
ment that the theory is unreasonable, it is not the case. The value of φ0 will not lead
to any contradictions, because any choice will lead to a consistent theory. Our choice
of this parameter says nothing about Nature, it only makes a statement about our
individual preference (Björk, Trifonov, Tsegaye and Söderholm 1998; Trifonov et al.
2000). Note as well, that the relative-phase eigenstates are maximally entangled states.
This has the consequence that the relative-phase operator has no classical correspon-
dence in the general case, not even for highly excited states.

Once again, the probability distribution function for the relative phase of a state
is described by the density matrix ˆ̺, as

P (N,φr, t) = 〈φ(N)
r | ˆ̺(t)|φ(N)

r 〉. (3.59)

However, for physical states (Barnett and Pegg 1989) (i.e., states for which finite
moments of the number operator are bounded) this expression will converge to a
simpler form involving a continuous probability density we shall write as

P (N,φ, t) = 〈φ(N)| ˆ̺(t)|φ(N)〉, (3.60)


3.1 Two-level atoms interacting with a quantum field 51

where the vectors |φ(N)〉 defined in (3.56) lie in the subspace HN with total number
of excitations N . In fact, this expression can be interpreted as a joint probability
distribution for the relative phase and the number of excitations. From it, we can
derive the distribution for the relative phase as

P (φ, t) =

∞
∑

N=0

P (N,φ, t), (3.61)

while

P (N, t) =

∫

2π

dφ P (N,φ, t) (3.62)

can be viewed as the probability distribution of having N excitations in the system.
These factorizations are an obvious consequence of the fact that the relative phase
and the excitation number are compatible.

For a general initial state as expressed in equation (3.43), and the evolution given
by (3.44) we have

P (N,φ, t) =
1

2π

∣

∣

∣

∣

∣

∣

D
∑

m′,m=0

QN−mAm CN
m′m(t)eim′φ

∣

∣

∣

∣

∣

∣

2

, (3.63)

and then we arrive at the total relative-phase probability distribution:

P (φ, t) =
1

2π

∞
∑

N=0

∣

∣

∣

∣

∣

∣

D
∑

m′,m=0

QN−mAm CN
m′m(t)eim′φ

∣

∣

∣

∣

∣

∣

2

. (3.64)

This is our basic and compact result which we use to analyze the evolution of the
phase properties of the Dicke model.

In figure 3.1 we have numerically evaluated this distribution P (φ, t) as a function
of φ and the rescaled adimensional time τ = gt, for the case when all the atoms are
initially unexcited and the field is in a coherent state with various values of the mean
number of photons n̄. In all the cases, when τ = 0 we have that CN

m′m(0) = δm′m and
therefore

P (φ, t = 0) =
1

2π

∞
∑

N=0

∣

∣

∣

∣

∣

D
∑

m=0

QN−mAm eimφ

∣

∣

∣

∣

∣

2

. (3.65)

In particular, when all the atoms are initially unexcited only the coefficient A0

survives and the previous expression reduces to

P (φ, t = 0) =
1

2π
. (3.66)


52 3 Relative phase in atom-field interactions

Fig. 3.1. Gray-level contour plot of the probability distribution P (φ, t) as a function of φ
and the rescaled time gt for the case of A = 5 atoms initially unexcited and the field in a
coherent state with the following values of the mean number of photons: a) n̄ = 1 (weak
field), b) n̄ = 50 (strong field), and n̄ = 5 (intermediate field).

This flat distribution reflects the fact that the random phase of the dipole in such
states induces a uniform distribution centred at φ0. At this respect, it is interesting to
notice that classically the Lorentz model at resonance predicts for the relative phase
values of ±π/2. It turns out that this is also a possible choice to fix the reference phase
φ0 in the quantum description. For simplicity, in all the graphics we have chosen φ0

as the origin 0.


3.1 Two-level atoms interacting with a quantum field 53

Two quite different behaviors are evident from these graphics. The first occurs in
the weak-field region (Heidmann et al. 1985; Kozierowski et al. 1992, 1990) when the
number of excitations in the system is much smaller than the number of the atoms,
N ≪ A. If, for simplicity, we assume that all the atoms are unexcited and the average
number of photons in the initially coherent field is small, say n̄ ∼ 1, then we can
retain only the dominant terms in (3.64), getting

P (φ, t) ≃ 1

2π
{1 + n̄[|C1

00(t)|2 + |C1
01(t)|2 + 2Re(C1

00(t)C
1
01(t)

∗
eiφ)]}e−n̄. (3.67)

We see that, due to the periodic temporal dependence of the terms CN
m′m(t), this dis-

tribution is oscillatory for all times, which is corroborated numerically in figure 3.1.a.
The second (and perhaps more interesting) case corresponds to the strong-field

region (Chumakov et al. 1994; Klimov and Chumakov 1995; Knight and Shore 1993;
l. Drobn and Jex 1993; Retamal et al. 1997), when the initial number of photons
is much larger than the number of atoms A ≪ N . Then, following the ideas of
(Chumakov et al. 1994; Klimov and Chumakov 1995; Retamal et al. 1997) one can
show that the coefficients CN

m′m(t) can be approximated, up to order A/
√
n̄, by

CN
m′m(t) ≃ dA

m′m(−ΩN t), (3.68)

where
ΩN = 2g

√

N −A/2 + 1/2, (3.69)

and dA
m′m are Wigner d functions,which are defined as the matrix elements for finite

rotations by operators from SU(2) group representations

dA
m′m(ϑ) = dA

mm′(ϑ) = 〈m′|eiϑŜx |m〉, (3.70)

where m,m′ = 0, 1, . . . , A. The point now is that essentially only one subspace of
dimension A+ 1 dominates the dynamics. Moreover, a calculation using the explicit
form of these d functions, gives

P (φ, t) =
1

2π

∞
∑

N=0

Q2
N

∣

∣

∣

∣

∣

A
∑

m=0

√

A!

(A−m)!m!
[tan(ΩN t/2)]meim(φ−π/2)

∣

∣

∣

∣

∣

2

[cos(ΩN t/2)]2A,

(3.71)
where we have assumed that all the atoms are initially unexcited. When A ≫ 1 and
when oscillations are well resolved, one can perform an expansion of the square root
getting

P (φ, t) =

√

A

2π

∑

N

φN (t) exp

[

−A
2

(φ− π/2 + δN )2
]

, (3.72)

where φN (t) is a function of time of complicated structure that accounts for the
collapses and revivals and that is of little interest for our purposes here, and δN =


54 3 Relative phase in atom-field interactions

arg[tan(ΩN t/2)]. Now, it is clear that, since δN takes only the values 0 and π, the
previous Gaussian distributions tend to have two peaks at φ = ±π/2, in agreement
with the classical expectations. The presence of collapses and revivals are evident in
figure 3.1.b, which confirms previous numerical and analytical evidence. The well-
known nearly time-independent behavior in the time windows between collapse and
revival is also clear. As we can see, the distribution tends to be randomized in the
evolution, although keeping these two peaks at ±π/2.

In the intermediate region, when N ∼ A, the behavior is more complex, as shown
in figure 3.1.c, and no analytical approximations are available.

For the particular case of the Jaynes-Cummings model one can diagonalize ex-
actly the Hamiltonian in each subspace HN , obtaining the well-known dressed states
(Cohen-Tannoudji et al. 1989), that turn to be trapping states
(Gea-Banacloche 1991); i.e., the atomic population 〈S3(t)〉 remains constant in spite of
the existence of both the radiation field and atomic transitions (Cirac and Sánchez-Soto
1990). These states play a fundamental role, so it seems interesting to analyze the
corresponding problem for the case of the Dicke model. In the strong-field limit one
can make the replacement a → α =

√
n̄ eiϑ and the interaction Dicke Hamiltonian

becomes proportional to the operator

Ĥcl =
(

eiϑŜ+ + e−iϑŜ−

)

, (3.73)

where the phase of the classical field has been chosen to coincide with the phase of
the initial coherent state of the field. The semiclassical atomic states are defined now
as eigenstates of Ĥcl taking this phase as zero:

2Ŝx|P 〉a = ΛP |P 〉a, (3.74)

with ΛP = A− 2P and P = 0, 1, . . . , A.
Following (Chumakov et al. 1994; Klimov and Chumakov 1995; Retamal et al.

1997), we shall call factorized states those states for which the initial field is taken
to be in a strong coherent state |α〉f and the atomic system is initially prepared in a
semiclassical atomic state |P 〉a. For such states, the total wavefunction of the system
can be approximately written as a product of its field and atomic parts

|Ψ(t)〉 ≃ |P (t)〉a ⊗ |α(t)〉f (3.75)

with

|P (t)〉a = exp

[

−i ΛP (Ŝ3 +A/2)

2
√

n̄−A/2 + 1/2
gt

]

|P 〉a

(3.76)

|α(t)〉f = exp

[

−iΛP

√

a†a−A/2 + 1/2 gt

]

|α〉f ,


3.1 Two-level atoms interacting with a quantum field 55

Fig. 3.2. Probability distribution function P (φ, t) as a function of φ and the rescaled time
gt for the case of a factorized state with A = 3. The atomic coherent state has ϑ = π/2 and
ϕ = 0 and the field state has n̄ = 20.

Fig. 3.3. Plots of 〈sin φ〉 versus gt for the same values of n̄ as in figure 3.1 (from top to
bottom).


56 3 Relative phase in atom-field interactions

and one can verify that they are also (approximately) trapping states.
For these states, one can find after a simple calculation,

P (φ, t) =
1

A+ 1

∣

∣

∣

∣

∣

∑

m

a〈m|P 〉aeimφ

∣

∣

∣

∣

∣

2

. (3.77)

The probability distribution is time independent due to the factorization (3.75). From
the arguments in (Chumakov et al. 1994; Klimov and Chumakov 1995; Retamal et al.
1997), one infers that this factorization holds up to times gt ∼

√
n̄ (which can be very

long times, in this limit) and with an accuracy in the coefficients of the order of A/
√
n̄.

Moreover, using the properties of the semiclassical atomic states and assuming
A≫ 1, one can replace the sum by an integral, obtaining finally

P (φ, t) ≃
√

A

2π
e−Aφ2/2; (3.78)

i.e., a Gaussian independent of time. In figure 3.2 we have plotted the probability
distribution obtained from a numerical computation of equation (3.64), showing this
quite remarkable behavior, except for the presence of very small (almost inappreciable)
oscillations superimposed.

To gain more physical insight in these behaviors, in figure 3.3 we have plotted the
evolution of the mean value of 〈sinφ〉 for various values of N , confirming the previous
physical discussion.

To conclude, let us consider the Dicke model in the large-detuning limit; which is
usually known as the dispersive limit. More specifically, we are in the case when

∆≫ g
√
n̄+ 1A. (3.79)

Then, following the procedure we shall develop in next chapter, the interaction Hamil-
tonian in equation (3.29) can be replaced by the effective Hamiltonian

Ĥeff = ∆ Ŝ3 + µ[Ŝ2
3 − 2(â†â+ 1)Ŝ3 − Ĉ], (3.80)

where

Ĉ =
A

2

(

A

2
+ 1

)

1̂1, µ =
µ2

∆
. (3.81)

The obvious advantage of this Hamiltonian is that it is diagonal and allows for a
compact analytical expression for the coefficients CN

m′m(t) as

CN
m′m(t) = δm′m exp (−it {∆(m−A/2) + µ [2(N −m) + 1] (m−A/2)

+ µ
[

C − (m−A/2)2
]})

. (3.82)

When the atoms are initially unexcited or excited (or, more generally, when Am =
δmk) and for any initial state of field, we have


3.1 Two-level atoms interacting with a quantum field 57

P (φ, t) =
1

2π
, (3.83)

for all the times.
For an arbitrary initial state of the atomic system and the field we get

P (φ, t) =
1

2π

∞
∑

N=0

∣

∣

∣

∣

∣

A
∑

m=0

QN−mAme
−ifN

m teimφ

∣

∣

∣

∣

∣

2

, (3.84)

with
fN

m = 2Nmµ+ [∆+ µ(2A+ 1)]m− 3µm2. (3.85)

Since (3.80) is quadratic in Ŝ3 and is, therefore, analogous to the Hamiltonian
quadratic in the number operator of a single-mode field propagating through a Kerr
medium, one could expect (Agarwal et al. 1997) that the evolution of coherent atomic
states in the dispersive limit of the Dicke model leads to the generation of Schrödinger
cat states. This superposition reaches the most pure form for initial number field states
(in particular, the vacuum state minimizes the atomic entropy (Klimov and Saavedra
1998)).

Fig. 3.4. Probability distribution function P (φ, t) as a function of φ for the time µt = π/6
for the case of an atomic coherent state (A = 5) with ϑ = π/2 and ϕ = 0 and a field coherent
state with n̄ = 10. The presence of the two humps corroborate the presence of a catlike state.

The situation with the relative-phase distribution is quite different. It is easy to
see, for example, that if the field is prepared initially in a number state |k〉a then
Qn = δkn and the relative phase distribution is flat. Nevertheless, for initial atomic


58 3 Relative phase in atom-field interactions

and field coherent states the relative-phase distribution splits for some special times
into several humps. These catlike states, according to (3.85), appear at times τ = µt =
π/6 ( mod 2π). To confirm analytically this behavior, we expand equation (3.84) when
initially we have strong coherent states for both field and atoms, with n̄ ≫ A ≫ 1.
By replacing once again the sum by integrals, one easily gets

P (φ, t = π/6λ) =

√

A

8π

{

e−[φ−πφn̄/(3µ)]2A/2 + e−[φ+π−πφn̄/(3µ)]2A/2
}

, (3.86)

where
φn̄ = 2n̄µ+A+ µ(2A+ 1), (3.87)

and all the phases must be understood mod(2π). The two separated Gaussians in-
dicates the presence of two humps and, therefore, the presence of catlike states. To
further confirm this, in figure 3.4 we have computed numerically the distribution
function P (φ, t = π/6λ) at the times predicted by the theory. The graphic clearly
demonstrates the presence of the two-component state, according to our previous
considerations.

3.2 Three-level atoms interacting with quantum fields

The natural way to continue is the analysis of the interaction of a three-level atom
with a two-mode electromagnetic field.

3.2.1 Three-level atom coupled to a two-mode field

We wish to explore in some detail the phase properties of a three-level system. To be
specific we shall consider a Λ configuration, as shown in figure 3.5, with energy levels
ω1 < ω2 < ω3 and with allowed dipole transitions 1 ↔ 3 and 2 ↔ 3, but not 1 ↔ 2.

The Hamiltonian for this system as

Ĥ = Ĥa + Ĥf + V̂ , (3.88)

where

Ĥa =
∑

i

ωiŜ
ii,

Ĥf = ωaâ
†â+ ωbb̂

†b̂,

V̂ = ga(âŜ13
+ + â†Ŝ13

− ) + gb(b̂Ŝ
23
+ + b̂†Ŝ23

− ). (3.89)

Here Ĥa describes the dynamics of the free atom and Ĥf represents the cavity modes
of frequency ωa and ωb, with annihilation operators â and b̂, respectively. Finally, in


3.2 Three-level atoms interacting with quantum fields 59

Fig. 3.5. The level scheme of a three-level Λ-type atom interacting with two single-mode
quantum fields, coupling the two ground states to a common excited atomic state.

the interaction term V̂ , written in the dipole and rotating-wave approximations, we
assume that the allowed transition 1 ↔ 3 couples (quasi)resonantly to the mode a
and the transition 2 ↔ 3 couples to the mode b with coupling constants ga and gb

that will be taken as real numbers.
The bare basis for the total system is |i〉a ⊗ |na, nb〉f , where |na, nb〉f , is the usual

two-mode Fock basis. However, one can check that the two excitation-number opera-
tors

N̂a = â†â− Ŝ11 + 1̂1, N̂b = b̂†b̂− Ŝ22 + 1̂1, (3.90)

are conserved quantities. In consequence, we can rewrite the Hamiltonian (3.88) as

Ĥ = Ĥ0 + Ĥint, (3.91)

with

Ĥ0 = ωaN̂a + ωbN̂b + (ω3 − ωa − ωb)1̂1,

(3.92)

Ĥint = −∆aŜ
11 −∆bŜ

22 + ga(âŜ13
+ + â†Ŝ13

− ) + gb(b̂Ŝ
23
+ + b̂†Ŝ23

− ).

The detunings are defined as

∆a = ω31 − ωa, ∆b = ω32 − ωb, (3.93)

with ωij = ωi − ωj . It is straightforward to check that

[Ĥ0, Ĥint] = 0. (3.94)


60 3 Relative phase in atom-field interactions

Therefore, both the free Hamiltonian Ĥ0 and the interaction Hamiltonian Ĥint are
constants of motion. Ĥ0 determines the total energy stored in the system, which
is conserved by the interaction. This allows us to factor out exp(−iĤ0t) from the
evolution operator and drop it altogether. Thus, the problem can be reduced to study
the restriction of Ĥint to each subspace H(Na,Nb) with fixed values of the pair of
excitation numbers (Na, Nb). In each one of these subspaces H(Na,Nb) there are three
basis vectors that can be written as

|i;na = Na − µi, nb = Nb − νi〉, (3.95)

where the values of µi and νi are defined as

(µ1, µ2, µ3) = (0, 1, 1), (ν1, ν2, ν3) = (1, 0, 1). (3.96)

Note that when Na = 1 and Nb = 0 or Na = 0 and Nb = 1 some states may
have negative photon occupation number and must be eliminated. In the subspace
H(Na,Nb), Ĥint is represented by the 3 × 3 matrix

Ĥ
(Na,Nb)
int =





0 gb

√
Nb ga

√
Na

gb

√
Nb −∆b 0

ga

√
Na 0 −∆a



 . (3.97)

Let us assume that at t = 0 the atomic wave function can be written as the
superposition

|Ψ(0)〉 =
∑

i

ci |i〉a ⊗ |αa, αb〉f (3.98)

with

|αa, αb〉 =

∞
∑

na,nb=0

Qna
Qnb

|na, nb〉f . (3.99)

At a later time t the state vector for the atom-field system in the interaction
picture can be expressed as

|Ψ(t)〉 =

∞
∑

Na,Nb=0

3
∑

i,j=1

QNa−µj
QNb−νj

cjU (Na,Nb)
ij (t) |i,Na − µi, Nb − νi〉,(3.100)

where U (Na,Nb)
ij (t) are the matrix elements of the evolution operator in the subspace

H(Na,Nb)

U (Na,Nb)
ij (t) = 〈i;Na − µi, Nb − νi| exp[−iĤ(Na,Nb)

int t]|j;Na − µj , Nb − νj〉, (3.101)

which can be calculated exactly as can be seen, e.g. in (Yoo and Eberly 1985). This
state describes completely the system evolution and will be the basis for our phase
analysis in the following.


3.2 Three-level atoms interacting with quantum fields 61

3.2.2 The role of atom-field relative phase

Our objetive is to describe the atom-field relative phase by resorting to a polar de-
composition of the corresponding complex amplitudes. To this end, let us define the
operators

X̂13
+ = â Ŝ13

+ , X̂13
z = Ŝ13

z ;

(3.102)

X̂23
+ = b̂ Ŝ23

+ , X̂23
z = Ŝ23

z .

These operators satisfy most of the usual su(3) commutation relations with one of
these recast as

[X̂13
+ , X̂23

− ] = −Ŷ 12
+ , (3.103)

where
Ŷ 12

+ = −âb̂†Ŝ12
+ . (3.104)

However, some of them must be modified in the following way

[X̂13
+ , X̂13

− ] = N̂a(1̂1 − 2Ŝ11 − Ŝ22),

[X̂23
+ , X̂23

− ] = N̂b(1̂1 − 2Ŝ22 − Ŝ11), (3.105)

[Ŷ 12
+ , Ŷ 12

− ] = N̂aN̂b(Ŝ
11 − Ŝ22),

which corresponds to a polynomial deformation of the algebra su(3).
For simplicity, let us focus first on the allowed transition 1 ↔ 3 and notice that

in every three-dimensional invariant subspace H(Na,Nb) the state |1;Na, Nb − 1〉 plays
the role of a vacuum state since

X̂13
− |1;Na, Nb − 1〉 = 0. (3.106)

In this subspace the operator X̂13
z is diagonal and we can work out again a polar

decomposition similar to (1.76), namely

X̂13
− =

√

X̂13
− X̂13

+ Ê13, (3.107)

where the operator
√

X̂13
− X̂13

+ is diagonal and

Ê13Ê13† = Ê13†Ê13 = Î ,

(3.108)

[Ê13, N̂a] = [Ê13, N̂b] = 0.


62 3 Relative phase in atom-field interactions

The first equation ensures that the operator Ê13 = eiφ̂13

, representing the exponential
of the relative phase between the field and the dipole 1 ↔ 3, is unitary. The second
one guarantees that we may study its restriction to each invariant subspace H(Na,Nb).

Much in the same way as we did in chapter 1 [see equation (1.77)], the operator
Ê13 solution of (3.107) can be expressed in H(Na,Nb) as

Ê13 = |1;Na, Nb − 1〉〈3;Na − 1, Nb − 1| − |3;Na − 1, Nb − 1〉〈1;Na, Nb − 1|
+ |2;Na − 1, Nb〉〈2;Na − 1, Nb|. (3.109)

As one would expect, Ê13 it acts as a ladder-like operator

Ê13|3;Na − 1, Nb − 1〉 = |1;Na, Nb − 1〉,
(3.110)

Ê13|2;Na − 1, Nb〉 = |2;Na − 1, Nb〉,
and thus has eigenvectors

|φ13
0 〉 = |2;Na − 1, Nb〉,

(3.111)

|φ13
± 〉 =

1√
2
(|3, Na − 1, Nb − 1〉 ± i|1;Na, Nb − 1〉),

while the eigenvalues of φ̂13 are 0 and ±π/2, respectively.
Obviously, a similar reasoning for the transition 2 ↔ 3 gives the corresponding

operator Ê23 as

Ê23 = |2;Na − 1, Nb〉〈3;Na − 1, Nb − 1| − |3;Na − 1, Nb − 1〉〈2;Na − 1, Nb|
+ |1;Na, Nb − 1〉〈1;Na, Nb − 1|, (3.112)

with eigenvectors

|φ23
0 〉 = |1;Na, Nb − 1〉,

(3.113)

|φ23
± 〉 =

1√
2
(|3, Na − 1, Nb − 1〉 ± i|2;Na − 1, Nb〉),

and the same eigenvalues as before. The states (3.111) and (3.113) are the basis for
our subsequent analysis of the dynamics of the relative phase.

3.2.3 Relative-phase distribution function

Once again, let us first focus on the relative phase between the field mode a and the
dipole transition 1 ↔ 3. According to (3.22) it seems natural to define the probability
distribution function of a state described by the density matrix ˆ̺(t) as


3.2 Three-level atoms interacting with quantum fields 63

P (Na, Nb, φ
13
r , t) = Tr[ˆ̺(t) |φ13

r 〉〈φ13
r |], (3.114)

where the vectors |φ13
r 〉 are given in equation (3.111) and the subscript r runs the

three possible eigenvalues 0, and ±π/2. This expression can be interpreted as a joint
probability distribution for the relative phase and the excitation operators N̂a and
N̂b. From this function, we can derive the distribution for the relative phase as the
marginal distribution

P (φ13
r , t) =

∞
∑

Na,Nb=0

P (Na, Nb, φ
13
r , t). (3.115)

For a general state as in equation (3.100), one has

P (Na, Nb, φ
13
r , t) = |〈φ13

r |Ψ(t)〉|2, (3.116)

which, through direct calculation, gives

P (φ13
0 , t) =

∞
∑

Na,Nb=0

∣

∣

∣

∣

∣

∣

3
∑

j=1

QNa−µj
QNb−νj

cj U (Na,Nb)
2j (t)

∣

∣

∣

∣

∣

∣

2

,

(3.117)

P (φ13
± , t) =

∞
∑

Na,Nb=0

∣

∣

∣

∣

∣

∣

3
∑

j=1

QNa−µj
QNb−νj

cj [U (Na,Nb)
3j (t) ± iU (Na,Nb)

1j (t)]

∣

∣

∣

∣

∣

∣

2

.

Much in the same way one also gets analogous results for the transition 2 ↔ 3:

P (φ23
0 , t) =

∞
∑

Na,Nb=0

∣

∣

∣

∣

∣

∣

3
∑

j=1

QNa−µj
QNb−νj

cj U (Na,Nb)
1j (t)

∣

∣

∣

∣

∣

∣

2

,

(3.118)

P (φ23
± , t) =

∞
∑

Na,Nb=0

∣

∣

∣

∣

∣

∣

3
∑

j=1

QNa−µj
QNb−νj

cj [U (Na,Nb)
3j (t) ± iU (Na,Nb)

2j (t)]

∣

∣

∣

∣

∣

∣

2

.

This is our basic and compact result to analyze the evolution of the relative phase.
We have numerically evaluated this distribution for the three allowed values of

the relative phase for the case when the atom is initially in the ground state |1〉 and
modes a and b are in a coherent state with a mean number of photons n̄a and n̄b,
respectively. For computational simplicity we have used the rescaled time

τ =
gat

2π
√
n̄a
, (3.119)


64 3 Relative phase in atom-field interactions

in all the plots and have assumed that ga = gb, which is not a serious restriction.
In figure 3.6 we have plotted a typical situation of a weak field, in which the number

of excitations in the system is small, say n̄a ∼ n̄b ∼ 1. The pattern shows an almost
oscillatory behavior, which can be easily understood if we retain only the two first
terms in the sums over Na and Nb and use the explicit form of the evolution operator
U . A relevant and general feature that is apparent from this figure is that the prob-
abilities associated with φij

+ and φij
− always oscillate out of phase, a point previously

demonstrated for the case of the Jaynes-Cummings model (Luis and Sánchez-Soto
1997).

Fig. 3.6. The probability distribution function for the six allowed values of the relative
phase as a function of the rescaled time τ in the case of a weak field with n̄a = n̄b = 1.

Perhaps more interesting is the case of strong-field dynamics, when the number
of excitations in the system is large and so n̄a or n̄b, or both, are large. In figure 3.7
we have plotted the relative-phase probabilities for n̄a = 50 and (a) n̄b = 0.5, and
(b) n̄b = 50 photons, with the atom initially in the ground state |1〉. When n̄b =
0.5, the distribution P (φ13

0 , t) (which is the probability of finding the atom in the
level |2〉) is almost negligible, while P (φ13

± , t) show collapses and revivals. One may
interpret this physically as follows: the transition 1 ↔ 3 is so intense due to stimulated
processes in mode a that there is no population transfer to level |2〉, which originates
a regular oscillation of the dipole 1 ↔ 3 with the corresponding collapses and revivals
in the relative phase. The well-known (nearly) time-independent behavior in the time
window between collapse and revival is also clear. The probability of finding the atom
in the level |1〉, P (φ23

0 , t), tends to be 1/2 (except at the revivals), which confirms
that the transition 1 ↔ 3 is almost saturated.


3.2 Three-level atoms interacting with quantum fields 65

Fig. 3.7. The probability distribution function for the six allowed values of the relative
phase as a function of the rescaled time τ in the strong-field limit with: a) n̄a = 50, n̄b = 0.5
and b) n̄a = 50, n̄b = 50.

When n̄b grows, the position of the collapses and revivals changes, according to
standard estimates (Yoo and Eberly 1985). When n̄a = n̄b = 50, P (φ13

0 , t) is centered
at 1/4, while P (φ23

0 , t) is centered at 1/2, showing that the populations tend to be
equidistributed because now the transition 2 ↔ 3 is almost saturated too.

Very interesting physical phenomena arise when one considers coherent superposi-
tions of atomic states, because it is then possible to cancel absorption or emission un-
der certain conditions, i.e., the atom is effectively transparent to the incident field even
in the presence of resonant transitions. A semiclassical analysis (Scully and Zubairy
1999), in which the fields are treated as c-numbers and described by the complex


66 3 Relative phase in atom-field interactions

Rabi frequencies Ωae
−iϑa and Ωbe

−iϑb (note that ϑa and ϑb are the ‘phases’ of the
respective fields), easily shows that when the initial atomic state is a superposition of
the two lower levels of the form

|Ψ(0)〉a =
1√
2
(|1〉 + eiϕ |2〉), (3.120)

coherent trapping occurs whenever

Ωa = Ωb, ϑa − ϑb − ϕ = ±π. (3.121)

In other words, when these conditions are fulfilled the population is trapped in the
lower states and there is no absorption.

To corroborate this behavior valid in the strong-field limit, let us note that, al-
though the transition 1 ↔ 2 is dynamically forbidden, one can still define phase
eigenvectors for it:

|φ12
0 〉 = |3, Na − 1, Nb − 1〉,

(3.122)

|φ12
± 〉 =

1√
2
(|2;Na − 1, Nb〉 ± i|1;Na, Nb − 1〉).

In figure 3.8 we have plotted the probabilities P (φ12
r , t) when the atom is initially in

a trapped state like (3.120) with n̄a = n̄b = 50. We see that P (φ12
0 , t), which is the

probability of finding the atom in the upper state, shows the remarkable behavior of
trapping, except for the presence of very small superimposed oscillations. Note also
that P (φ12

± , t) also shows the same kind of behavior.


3.2 Three-level atoms interacting with quantum fields 67

Fig. 3.8. The probability distribution function for the allowed values of the relative phase
φ̂12 as a function of the rescaled time τ for a trapped state with n̄a = n̄b = 50.


4 Small rotations and effective Hamiltonians

In most nonlinear optical phenomena, the fields and the atomic transitions are usually
far from resonance (Schenzle 1981). In fact, no appreciable population redistribution
is brought about the irradiation even of intense fields. The atomic polarization is then
a fast variable controlled by the slow motion of the electromagnetic field amplitudes
and follows its evolution adiabatically. Under this assumption we can eliminate the
atomic degrees of freedom and are left with a small number of equations for the
fields alone. These equations appear as suffering a nonlinear field-field interaction and
can consequently be reinterpreted as the Heisenberg equations of motion for the field
operators under the dynamics of an effective nonlinear Hamiltonian.

However, this adiabatic elimination (that was first used to study the paramet-
ric oscillator (Graham 1968; Graham and Haken 1968)) presents some drawbacks.
First, it does not provide a general prescription for finding effective Hamiltonians,
since the particular details strongly depend on the model considered. Second, it could
become very cumbersome, and explicit but enormously complicated expressions for
the different orders of approximation can be found in many original publications
(Bloembergen 1990; Shen 1985). Third, and even worse, the procedure is not uniquely
defined: depending on the term eliminated, the final Hamiltonian could be different
(Klimov et al. 1999; Lugiato et al. 1990; Puri and Bullough 1988).

In consequence, it seems pertinent to find a setting to support this usual ap-
proach of effective Hamiltonians. To this end, we first note that most of the effective
Hamiltonians in quantum optics contain cubic or higher terms in creation and an-
nihilation operators. Among others, typical examples are kth harmonic generation,
k-wave mixing, and generalized Dicke models. The key point for our purposes is the
recent observation that the common mathematical structure underlying all these cases
is a polynomial deformation of su(2), which arises as the dynamical symmetry algebra
of the corresponding Hamiltonian. This nonlinear algebra has recently found an im-
portant place in quantum optics (Debergh 1997; Delgado et al. 2000; Karassiov 1992,
1994; Sunilkumar et al. 2000) because it allows us to handle the problems in very
close analogy with the usual treatment of an angular momentum.


70 4 Small rotations and effective Hamiltonians

In (Klimov and Sánchez-Soto 2000) a Lie-like was devised method (Steinberg
1987) that allows one to get approximately effective diagonal Hamiltonians, provided
the existence of this nonlinear su(2) dynamical algebra.

Since the su(3) algebra is the natural extension of su(2) three-level systems, when
three-level systems interact with quantum fields, a nonlinear deformation of su(3)
naturally emerges. In this chapter we apply our method to some nontrivial examples:
a collection of two-level atoms dispersively interacting with a quantum field in a
cavity; and Ξ and Λ configurations of three-level systems interacting with fields under
different resonance conditions. Finally, Hamiltonians for multilevel systems under N -
photon resonance conditions.

4.1 Effective Hamiltonians for nonlinear su(2) dynamics

4.1.1 Motivation of the method

To keep the discussion as self-contained as possible and to introduce the physical ideas
underlying the method, let us start with the very simple example of a particle of spin
s in a magnetic field. The Hamiltonian for this system has the following form

Ĥ = ωŜ3 + g(Ŝ+ + Ŝ−), (4.1)

where g is the coupling constant and the operators Ŝ3, Ŝ+, and Ŝ− constitute a (2s+
1)-dimensional representation of the su(2) algebra, obeying the usual commutation
relations.

The Hamiltonian (4.1) belongs to the class of the so-called linear Hamiltonians
(Peřinova and Lukš 1994) and admits an exact solution. A convenient way of working
out the solution is to apply the rotation

Û = exp
[

α(Ŝ+ − Ŝ−)
]

, (4.2)

and recalling that eÂB̂e−Â = B̂+[Â, B̂]+ 1
2! [Â, [Â, B̂]]+ . . ., the rotated Hamiltonian,

which is unitarily equivalent to the original one, becomes

Ĥ = ÛĤÛ† = [ω cos(2α)+2g sin(2α)]Ŝ3 +
1

2
[2g cos(2α)−ω sin(2α)](Ŝ+ + Ŝ−). (4.3)

Now, the central idea is to choose the parameter α so as to cancel the nondiagonal
terms appearing in equation (4.3). This can be accomplished by taking

tan(2α) =
2g

ω
, (4.4)

and the transformed Hamiltonian reduces then to


4.1 Effective Hamiltonians for nonlinear su(2) dynamics 71

Ĥeff = ω

√

1 +
4g2

ω2
Ŝ3. (4.5)

Since this effective Hamiltonian is diagonal in the angular momentum basis, the dy-
namical problem is completely solved. The crucial observation is that when g ≪ ω
we can approximate equation (4.4) by α ≃ g/ω, and then equation (4.2) can be
substituted by

Û ≃ exp
[ g

ω
(Ŝ+ − Ŝ−)

]

. (4.6)

This small rotation approximately (i.e., up to second-order terms in g/ω) diagonalizes
the Hamiltonian (4.1), giving rise to

Ĥeff = ÛĤÛ† ≃
(

ω + 2
g2

ω

)

Ŝ3 (4.7)

which obviously coincides with the exact solution (4.5) after expanding in a series
of g2/ω2. A direct application of the standard time-independent perturbation theory
(Cohen-Tannoudji et al. 1992) to equation (4.1) leads immediately to the same eigen-
values and eigenstates than (4.7) in the same order of approximation. However, we
stress that our method is fully operatorial and avoids the tedious work of computing
the successive corrections as sums over all the accessible states.

4.1.2 Nonlinear small rotations

Having in mind the previous simple example, let us go one step further by treating
the more general case of a system that admits some integrals of motion N̂j and whose
interaction Hamiltonian can be written as

Ĥint = ∆ X̂3 + g(X̂+ + X̂−), (4.8)

where g is the coupling constant and ∆ is a parameter usually representing a detuning
between frequencies of different subsystems (although it is not strictly necessary). The
operators X̂± and X̂3 maintain the first commutation relation of su(2), [X̂3, X̂±] =
±X̂±, but the second one is modified in the following way

[X̂+, X̂−] = P (X̂3), (4.9)

where P (X̂3) is an arbitrary polynomial function of the diagonal operator X̂3 with
coefficients perhaps depending on the integrals of motion N̂j . These commutation
relations correspond once again to a polynomial deformation of su(2), as the ones
treated in chapter 2.

Now, suppose that for some physical reasons (depending on the particular model
under consideration) the condition


72 4 Small rotations and effective Hamiltonians

ε =
g

∆
≪ 1 (4.10)

is fulfilled. Then, it is clear that (4.8) is almost diagonal in the basis that diagonalizes
X̂3. In fact, a standard perturbation analysis immediately shows that the first-order
corrections introduced by the nondiagonal part g(X̂+ + X̂−) to the eigenvalues of X̂3

vanish and those of second order are proportional to ε. Thus, we apply the following
unitary transformation to (4.8) (which, in fact, is a small nonlinear rotation)

Û = exp[ε(X̂+ − X̂−)]. (4.11)

After some calculations we get, up to order ε2, the effective Hamiltonian

Ĥeff = ∆ X̂3 +
g2

∆
P (X̂3). (4.12)

Then, the evolution (as well as the spectral) problem is completely solved in this
approximation. Besides the advantage of having the effective Hamiltonian expressed
in an operatorial form, the method has the virtue of generality, since it is valid for
any model whose Hamiltonian could be written down in terms of the generators of a
polynomial deformation of su(2).

Our technique also provides a valuable tool for obtaining corrections to the eigen-
states of (4.8). Indeed, it is easy to realize that these eigenstates can be approximated
by

|Ψm〉 = Û†|m〉, (4.13)

where |m〉 denotes an eigenstate of X̂3 and Û is the corresponding small rotation.
Since U and |m〉 do not depend on time, the operator Û can be applied to |m〉 as an
expansion in ε. For example, the eigenstate |Ψm〉 up to order ε2 takes on the form

|Ψm〉 =

[

1 − ε (X̂+ − X̂−) − ε2

2
(1 + 2X̂+X̂− − X̂2

+ − X̂2
−)

]

|m〉. (4.14)

This representation is especially advantageous when the state space of the model is
a representation space of the deformed su(2) algebra that is constructed in the usual
way by the action of the raising operator X̂+; i.e., |m〉 ∝ X̂m

+ |0〉, where |0〉 is a lowest

weigth vector fulfilling the standard condition X̂−|0〉 = 0.
This procedure shows that we can adiabatically eliminate all nonresonant transi-

tions and work with an effective Hamiltonian containing only (quasi) resonant tran-
sitions. The effect of nonresonant terms reduces to a dynamical Stark shift (which
can have a quite complicated form). Obviously, transformations generating effective
Hamiltonians also change eigenfunctions, but the corrections are of order ε and do
not depend on time. Such corrections correspond to low-amplitude transitions that
take place in the case of nonresonant interactions. We shall elaborate on these topics
in the next sections.


4.2 Effective Hamiltonians for nonlinear su(3) dynamics 73

4.1.3 Dispersive limit of the Dicke model

As a relevant example, let us apply our method to the Dicke model, describing the
interaction of a single-mode field of frequency ω with a collection of A identical two-
level atoms with transition frequency ω0. The model Hamiltonian is given in equation
(2.2).

We assume now that the dispersive limit holds (Brune et al. 1996); i.e.,

|∆| ≫ Aλ
√
n̄+ 1, (4.15)

where n̄ is the average number of photons in the field. If we apply our method, con-
sidering the deformed su(2) operators in equation (2.18), the small nonlinear rotation
(4.11) transforms the interaction Hamiltonian (2.2) into

Ĥeff = ∆ Ŝ3 +
λ2

∆
[Ŝ2

3 − 2(â†â+ 1)Ŝ3 − Ĉ], (4.16)

where

Ĉ =
A

2

(

A

2
+ 1

)

1̂1, µ =
λ2

∆
. (4.17)

This effective Hamiltonian (4.16) was previously obtained in (Agarwal et al. 1997) by
quite a different method (see also (Klimov and Saavedra 1998)).

4.2 Effective Hamiltonians for nonlinear su(3) dynamics

4.2.1 Three-level systems interacting with quantum fields

The method of small rotations can be applied to more complicated systems. In this
section we focus on Hamiltonians that can be represented in terms of the su(3) al-
gebra. This structure naturally arises when dealing with systems with three relevant
levels. In this case three possible configurations (Ξ, V, and Λ) are admissible [see
(Yoo and Eberly 1985) for details].

For definiteness, we shall consider the interaction of A identical three-level systems
in a cascade or Ξ configuration (i.e., with associated energies E1 < E2 < E3 and
allowed dipole transitions 1 ↔ 2 and 2 ↔ 3, but not the 1 ↔ 3) interacting with a
single-mode quantum field of frequency ω. The Hamiltonian of this model is

ĤΞ = Ĥf + Ĥa + Ĥint, (4.18)

with


74 4 Small rotations and effective Hamiltonians

Ĥf = ωâ†â,

Ĥa = E1Ŝ
11 + E2Ŝ

22 + E3Ŝ
33, (4.19)

Ĥint = g12(âŜ
12
+ + â†Ŝ12

− ) + g23(âŜ
23
+ + â†Ŝ23

− ).

It is natural also to introduce the deformed su(3) algebra as

X̂11 = Ŝ11, X̂22 = Ŝ22, X̂33 = Ŝ33,

(4.20)

X̂12
+ = âŜ12

+ , X̂23
+ = âŜ23

+ .

Then the Hamiltonian (4.18) can be recast as ĤΞ = Ĥ0 + Ĥint, with

Ĥ0 = ωN̂Ξ ,

(4.21)

Ĥint = −∆12 X̂
11 +∆23 X̂

33 + g12(X̂
12
+ + X̂12

− ) + g23(X̂
23
+ + X̂23

− ),

where we have used the conserved excitation number N̂Ξ = â†â+ Ŝ33 − Ŝ11 and the
detunings are

∆12 = E2 − E1 − ω, ∆23 = E3 − E2 − ω. (4.22)

The operators X̂ij satisfy the usual su(3) commutation relations (1.75), with some of
them recast as

[X̂12
+ , X̂23

+ ] = −Ŷ 13
+ , [X̂12

− , X̂23
− ] = Ŷ 13

− , [X̂12
+ , X̂23

− ] = 0, (4.23)

where
Ŷ 13

+ = â2Ŝ13
+ . (4.24)

However, we have to modify some of them in the following way:

[X̂ij
+ , X̂

ij
− ] = P (X̂ii, X̂jj), [Ŷ ij

+ , Ŷ ij
− ] = Q(X̂ii, X̂jj), (4.25)

where P (X̂ii, X̂kk) and Q(X̂ii, X̂kk) are polynomials of the diagonal operators X̂ii

(i = 1, 2, 3) and define, then, a polynomial deformation of su(3).

Effect of a far-off resonant level

The dynamics generated by su(3) is obviously richer than that of su(2), since a greater
number of physical degrees of freedom are now available. To see how our method
works, let us assume that one of the transitions, say the 2 ↔ 3, is (quasi) resonant
with the field; i.e.,

|∆23| ≪ Ag23
√
n̄+ 1, (4.26)


4.2 Effective Hamiltonians for nonlinear su(3) dynamics 75

while the transition 1 ↔ 2 is far-off resonant

|∆12| ≫ Ag12
√
n̄+ 1. (4.27)

It is clear from our previous analysis that the small nonlinear rotation

Û12 = exp[ε12(X̂
12
+ − X̂12

− )], (4.28)

with ε12 = g12/∆12 ≪ 1, eliminates the interaction term g12(X̂
12
+ +X̂12

− ), representing
the nonresonant transition 1 ↔ 2. In this way, we obtain the effective Hamiltonian

Ĥeff = ∆12 X̂
11 +∆23 X̂

33 + g23(X̂
23
+ + X̂23

− ) − g12g23
∆12

(Ŷ 13
+ + Ŷ 13

− )

+
g2
12

∆12
P (X̂11, X̂22) +

g12g23
∆12

([X̂12
+ , X̂23

− ] + [X̂23
+ , X̂12

− ]). (4.29)

The remarkable point is that by eliminating the transition 1 ↔ 2, we have generated
an effective transition 1 ↔ 3 (represented by the operators Ŷ 13

± ), which was absent
in the initial Hamiltonian. Nevertheless, this transition is also nonresonant due to
conditions (4.26) and (4.27) and, accordingly, can be eliminated by the following
transformation

Û13 = exp[ε13(Ŷ
13
+ − Ŷ 13

− )], (4.30)

where the parameter ε13 must fulfill

ε13 =
g12g23

∆12(∆12 +∆23)
≪ 1. (4.31)

In this particular case the polynomial function P (X̂ii, X̂jj) is

P (X̂11, X̂22) = Ŝ12
+ Ŝ12

− + â†â(Ŝ22 − Ŝ11). (4.32)

If initially level 1 is unpopulated, we have Ŝ11 = 0 (which will be conserved, since
there are no transitions to the level 1) and

P (X̂11, X̂22) = Ŝ22(â†â+ 1). (4.33)

In consequence, the effective Hamiltonian, taking into account the existence of a far-off
resonant level, has the form

Ĥeff = ∆23 Ŝ
33 + g23(âŜ

23
+ + â†Ŝ23

− ) +
g2
12

∆12
Ŝ22(â†â+ 1). (4.34)

This means that the far-lying level produces a mark in the system in the form of a
dynamical Stark-shift term. Due to this Stark shift, the initially nonresonant tran-
sition 2 ↔ 3 becomes resonant in a subspace with some fixed photon number. This
opens the possibility of separating n-photon field states from an initial coherent state
interacting with a collection of three-level atoms just by choosing a suitable relation
between the detunings and the interaction constants, and projecting to the second
level at appropriate moments.


76 4 Small rotations and effective Hamiltonians

Two-photon resonance

Let us now envisage the different situation in which the two-photon resonance condi-
tion between levels 1 and 3 is fulfilled; i.e., E3−E1 = 2ω. This means that∆12 = −∆23

and the transition generated by the operators Ŷ 13
± cannot be removed. However, the

term g23(X̂
23
+ + X̂23

− ), which generates (nonresonant) transitions between levels 2 and

3, can be eliminated by a transformation Û23 analogous to (4.28) with ε23 ≪ 1. The
transformed Hamiltonian becomes then

Ĥeff = −∆12 (X̂11 + X̂33) − g12g23
∆12

(Ŷ 13
+ + Ŷ 13

− )

+
g2
12

∆12
P (X̂11, X̂22) − g2

23

∆12
P (X̂22, X̂33), (4.35)

where P (X̂11, X̂22) and P (X̂22, X̂33) are defined according to equation (4.32). Finally,
if we further impose the absence of initial population in level 2, we obtain

Ĥeff =
g12g23
∆12

(â2Ŝ13
+ + â†2Ŝ13

− )

+ (Ŝ13
3 +A/2)

[

(g2
23/∆12 − g2

12/∆12)â
†â+ g2

23/∆12

]

+A
g2
12

∆12
â†â, (4.36)

which is the effective two-photon Dicke Hamiltonian including the dynamical Stark
shift obtained in (Puri and Bullough 1988) [see also (Klimov et al. 1999)].

4.2.2 The dispersive limit of the Λ configuration

Let us consider for a moment the case of the Λ configuration, in which the allowed
dipole transitions are now 1 ↔ 3 and 2 ↔ 3, but not the 1 ↔ 2. The Hamiltonian
governing the evolution is still of the form (4.18), but now with

Ĥint = g13(âŜ
13
+ + â†Ŝ13

− ) + g23(âŜ
23
+ + â†Ŝ23

− ). (4.37)

By using the integral of motion N̂Λ = â†â+ Ŝ33 we can rewrite

Ĥint = −∆31 X̂
11 −∆32 X̂

22 + g13(X̂
13
+ + X̂13

− ) + g23(X̂
23
+ + X̂23

− ), (4.38)

where, as in equation (4.20), we have introduced the deformed su(3) operators as

X̂13
+ = âŜ13

+ , X̂23
+ = âŜ23

+ . (4.39)

These deformed generators satisfy a set of commutation relations similar to (4.23),
but instead of (4.24) we must use


4.3 Effective Hamiltonians for nonlinear su(d) dynamics 77

Ŷ 12
+ = (Ŝ33 − â†â), (4.40)

which is the mathematical reason for the well-known different behaviours exhibited
by Ξ and Λ configurations.

Let us focus on the dispersive regime, when

|∆13| ≫ Ag13
√
n̄+ 1, |∆23| ≫ Ag23

√
n̄+ 1. (4.41)

Then a couple of small rotations eliminate the far-off resonant transitions 1 ↔ 3 and
2 ↔ 3, obtaining

Ĥeff = −∆31 Ŝ
11 −∆32 Ŝ

22

+
g2
13

∆13
[(Ŝ11 + 1)Ŝ33 + â†â(Ŝ33 − Ŝ11)] +

g2
23

∆23
[(Ŝ22 + 1)Ŝ33 + â†â(Ŝ33 − Ŝ22)]

+
g13g23
∆31

(Ŝ12
+ + Ŝ12

− )(Ŝ33 − â†â). (4.42)

Note that both â†â and Ŝ33 are now integrals of motion. The two first terms cor-
respond to trivial free atomic dynamics. The next two terms represent the standard
dynamical Stark shift. Finally, the last term describes an effective interaction between
levels 1 and 2. The remarkable point is that there is a population transfer (and not
only phase transfer, as it could be expected from a dispersive interaction) between
these two levels without exchange of photons. The intensity of the transition 1 ↔ 2
depends on the difference between the population of level 3 and the photon number.
Thus, no population transfer between levels 1 and 2 will occur in the sector where
the number of photons is exactly equal to the initial population of the level 3. It is
easy to observe that the (effective) transitions 1 ↔ 2 are stronger when ∆31 = ∆32;
i.e., when the levels 1 and 2 have the same energy (Zeeman-like systems).

4.3 Effective Hamiltonians for nonlinear su(d) dynamics

4.3.1 Multilevel systems interacting with a quantum field

Having demonstrated the role played by nonlinear algebras in the systematic con-
struction of effective Hamiltonians, we would like to pursue here a natural extension
to multilevel systems. Specifically, we are interested in considering Hamiltonians that
can be represented in terms of the su(d) algebra, which naturally arises when describ-
ing systems with d levels. Obviously, the evolution of a collection of A identical d-level
systems (for definiteness, we assume a cascade configuration, such that Ei < Ej for
i < j) interacting with a single-mode quantum field can be modeled by a Hamiltonian
as in equation (4.18) with


78 4 Small rotations and effective Hamiltonians

Ĥf = ω â†â,

Ĥa =

d
∑

i=1

Ei Ŝ
ii, (4.43)

Ĥint =

d−1
∑

i=1

gi(âŜ
ii+1
+ + â†Ŝii+1

− ).

where Ŝii (i = 1, . . . , d) are population operators of the ith energy level, and Ŝij
±

describe transitions between levels i and j. The operators Ŝij form the u(d) algebra
and satisfy the commutation relations

[Ŝij , Ŝkl] = δjkŜ
il − δilŜ

kj . (4.44)

By introducing inversion-like operators

Ŝii+1
3 =

1

2
(Ŝi+1i+1 − Ŝii), (4.45)

then (Ŝij
± , Ŝ

ii+1
3 ) turn out to be the su(d) algebra.

The Hamiltonian (4.43) admits the integral of motion

N̂ = â†â+
d−1
∑

i=1

µi Ŝ
ii+1
3 , (4.46)

with µi = i(d− i). We also introduce the detunings by

∆j = Ej − E1 − (j − 1) ω, (4.47)

and assume that ∆j satisfy the following resonant condition

∆d = 0, (4.48)

which means that the field in a (d−1)-photon resonance with the atomic system; i.e.,
Ed − E1 = (d − 1)ω. Thus, the Hamiltonian (4.43) can be recast as Ĥ = Ĥ0 + Ĥint,
with

Ĥ0 = ωN̂ +A E

(4.49)

Ĥint = ĥ0 + V̂ ,

where E = (Ed + E1)/2 and


4.3 Effective Hamiltonians for nonlinear su(d) dynamics 79

ĥ0 =

d
∑

i=1

∆i Ŝ
ii, V̂ =

d−1
∑

i=1

gi(âŜ
ii+1
+ + â†Ŝii+1

− ). (4.50)

From our previous experience it is clear that the operators

X̂ii = Ŝii, X̂ij
+ = âŜij

+ , X̂ij
− = â†Ŝij

− , (4.51)

form a polynomial deformation of su(d). In consequence, and according to our general
scheme, we can introduce the transformation

Û = exp

[

d−1
∑

i=1

εi(X̂
ii+1
+ − X̂ii+1

− )

]

, (4.52)

where
εi =

gi

∆i+1 −∆i
(4.53)

are assumed to be small numbers, εi ≪ 1, which means that the transitions are far
from the one-photon resonance (∆i+1 − ∆i = Ei+1 − Ei − ω ≫ gi). Thus, all one-
photon transitions are eliminated by (4.52) and the transformed Hamiltonian takes
the form

Ĥ
(1)
eff = ĥ0 + ĥd + ĥnd

+
d−2
∑

ℓ=1

ℓ

(ℓ+ 1)!

d−ℓ−1
∑

i=1

λ
(ℓ+1)
i

(

âℓ+1 Ŝii+ℓ+1
+ + â†ℓ+1 Ŝii+ℓ+1

−

)

. (4.54)

Here, the effective interaction constants λ
(n)
i can be obtained from the recurrence

relation
λ

(n+1)
i = εi+nλ

(n)
i − εiλ

(n)
i+1, (4.55)

with the initial term λ
(1)
i = gi. It is easy to see that λ

(n+1)
i ≪ λ

(n)
i .

The piece hd contains only diagonal terms in the atomic operators and depends
on the integral of motion N̂ (or, equivalently, depends only on the photon-number
operator â†â). This operator hd appears naturally represented as an expansion in the
small parameter εi whose first term is

ĥd =
d−1
∑

i=1

giεi[â
†â(Ŝi+1i+1 − Ŝii) + (Ŝii + 1)Ŝi+1i+1]. (4.56)

The essential point is that, given its structure, this diagonal part cannot be removed
from the effective Hamiltonian (4.54). On the contrary, the operator ĥnd contains only
nondiagonal terms that can be eliminated by rotations of the type (4.52) unless some
specific resonance conditions are fulfilled. In this respect, let us note that the price we
pay for eliminating one-photon transitions is the generation of all possible k-photon
transitions (k = 2, ..., d− 1).


80 4 Small rotations and effective Hamiltonians

4.3.2 Three-photon resonance

Let us consider the particular case of four-level systems (d = 4) and suppose that there
are no transitions in one- and two-photon resonance with the field. After eliminating
one-photon transitions the transformed Hamiltonian (4.54) has the form

Ĥ
(1)
eff = ĥ0 + ĥd + ĥnd

+
1

3
λ

(3)
1

(

â3Ŝ14
+ + â†3Ŝ14

−

)

+
1

2

2
∑

i=1

λ
(2)
i

(

â2Ŝii+2
+ + â†2Ŝii+2

−

)

, (4.57)

where the interaction constants are defined, according to (4.53) and (4.55), as

ε1 =
g1
∆2

, ε2 =
g2

∆3 −∆2
, ε3 = − g3

∆3
,

(4.58)

λ
(2)
1 = g1g2

2∆2 −∆3

∆2(∆3 −∆2)
, λ

(2)
2 = g2g3

2∆3 −∆2

∆3(∆2 −∆3)
, λ

(3)
1 =

3g1g2g3
∆3∆2

,

and the resonance condition ∆4 = 0 (that is, three-photon resonance E4 − E1 = 3ω)
has been imposed.

According to the general scheme, the term representing two-photon transitions in
(4.57) can be removed using a transformation analogous to (4.52):

Û2 = exp

[

1

2

2
∑

i=1

α
(2)
i (â2Ŝii+2

+ − â†2Ŝii+2
− )

]

, (4.59)

where

α
(2)
i =

λ2
i

∆i+2 −∆i
(4.60)

is a small parameter because there are no resonant two-photon transitions (∆i+2 −
∆i = Ei+2−Ei−2ω ≫ λ

(2)
i ) and thus α

(2)
j ≪ εj . It is worth noting that the transfor-

mation (4.59) does not introduce new terms of order ε2 to the effective Hamiltonian.
The diagonal part in (4.57) is

ĥd =
3

∑

i=1

giεi

[

â†â(Ŝi+1i+1 − Ŝii) + (Ŝii + 1)Ŝi+1i+1
]

, (4.61)

while the nondiagonal term deserves a more careful analysis. Its explicit form is

ĥnd =
1

2
(ε1g3 + ε3g1)(Ŝ

12
+ Ŝ34

− + Ŝ34
+ Ŝ12

− ) +
1

2
(ε1g2 + ε2g1)(Ŝ

12
+ Ŝ23

− + Ŝ23
+ Ŝ12

− ). (4.62)


4.3 Effective Hamiltonians for nonlinear su(d) dynamics 81

It is clear that the first term in the above expression describes a resonant dipole-
dipole interaction, under the condition ∆2 = −∆3. On the other hand, the second
term describes a resonant interaction whenever 2∆2 = ∆3, which is incompatible
with the pervious one and the absence of one- and two-photon resonances. If no one
of these conditions are fulfilled, the term ĥnd can be eliminated by the transformation

Û
(2)
2 = exp





1

2

3
∑

i,j=1

βij(Ŝ
ii+1
+ Ŝjj+1

− − Ŝjj+1
+ Ŝii+1

− )



 , (4.63)

where
βij =

εigj

∆i+1 −∆i +∆j −∆j+1
. (4.64)

Then, since Ŝ11 + Ŝ22 + Ŝ33 + Ŝ44 = A and imposing the condition of the absence
of initial population in levels 2 and 3, we obtain the effective Hamiltonian describing
three-photon resonant transitions

Ĥ
(2)
eff =

g1g2g3
∆2∆3

(â3Ŝ14
+ + â†3Ŝ14

− )

− (Ŝ14
3 +A/2)[â†â(g2

1/∆2 − g2
3/∆3) + g2

3/∆3] +A
g2
1

∆2
â†â. (4.65)

The effective three-photon Hamiltonian (4.65) contains a dynamical Stark shift
similar to that appearing in the two-photon case (4.36). Nevertheless, in the two-
photon case the interaction term and the Stark shift are of the same order of magni-
tude, while in the three-photon case the interaction term is one order of magnitude
less than the Stark shift. This would lead to essential differences in the evolution of
some observables.

4.3.3 Multimode fields and resonance conditions

The formalism developed can also be used to treat the interaction of multifrequency
fields with atomic systems. This type of interaction is much richer because some
additional resonance conditions can be satisfied. To this end, let us consider a four-
level system in a Ξ configuration interacting with two field modes of frequencies ωa

and ωb, and annihilation operators â and b̂, respectively. The Hamiltonian describing
this interaction is still of the form (4.18) with

Ĥf = ωa â
†â+ ωb b̂

†b̂,

Ĥa =

4
∑

i=1

Ei Ŝ
ii, (4.66)

Ĥint =

3
∑

i=1

[

(gaiâ+ gbib̂)Ŝ
ii+1
+ + (gaiâ

† + gbib̂
†)Ŝii+1

−

]

,


82 4 Small rotations and effective Hamiltonians

where gai and gbi are the corresponding coupling constants. We assume that the
following resonance conditions are satisfied:

E4 − E1 = 3ωb, E3 − E1 = 2ωb, (4.67)

while all one-photon transitions are out of resonance. In such a case, the Hamiltonian
(4.66) may describe, for example, a fifth-order process involving the absorption of
three photons from one field and the stimulated emission of two photons of different
frequency. The effective Hamiltonian describing explicitly these transitions can be
obtained according to the general method. Now the integral of motion (4.46) takes
the form

N̂ = â†â+ b̂†b̂+

3
∑

i=1

µiŜ
ii+1
3 , (4.68)

where µi = i(4 − i), and the Hamiltonian can be recast also as in (4.49) with Ĥint =

ĥ0 + V̂a + V̂b, where

ĥ0 = δ b̂†b̂+
4

∑

i=1

∆iŜ
ii,

(4.69)

V̂a =
3

∑

i=1

gai(X̂
ii+1
a+ + X̂ii+1

a− ), V̂b =
3

∑

i=1

gbi(X̂
ii+1
b+ + X̂ii+1

b− ).

Here the polynomial deformation is defined by

X̂ij
a+ = âŜij

+ , X̂ij
a− = â†Ŝij

− ; X̂ij
b+ = b̂Ŝij

+ , X̂ij
b− = b̂†Ŝij

− , (4.70)

and δ = ωb − ωa.
Now we can eliminate all the one-photon nonresonant transitions generated by

means of two transformations (one for mode â the other for mode b̂) identical to
(4.52). The transformed Hamiltonian (up to order 1/∆2) is

Ĥ
(1)
eff = ĥ0 + ĥ

(a)
d + ĥ

(b)
d +

1

2
λ

(2)
1

(

â2Ŝ13
+ + â†2Ŝ13

−

)

+
1

3
λ

(3)
1

(

b̂3Ŝ14
+ + b̂†3Ŝ14

−

)

+ ξ
(ab)
2

(

âb̂Ŝ24
+ + â†b̂†Ŝ24

−

)

, (4.71)

donde ĥ
(a)
d and ĥ

(b)
d are defined according to (5.49) for both modes and the coupling

constants λ
(2)
1 and λ

(3)
1 are given by (4.58). The constant ξ

(ab)
2 is

ξ
(ab)
2 =

ga3gb2

∆4 −∆3
− gb3ga2

∆3 −∆2
. (4.72)


4.3 Effective Hamiltonians for nonlinear su(d) dynamics 83

This effective Hamiltonian deserves some comments. First of all, when the reso-
nance condition

E4 − E2 = ωa + ωb (4.73)

is fulfilled (which is compatible with the two- and three-photon resonance conditions),
then the last term in the Hamiltonian describes resonant transitions between levels 2
and 4, as a result of the simultaneous absorption and emission of quanta from both
modes, and take place only if levels 2 and 4 are populated. In absence of such an
additional resonance condition, the Hamiltonian (4.71) describes two simultaneous

and competing processes: a first-order process (λ
(2)
1 ∼ 1/∆) involving two-photon

transitions between levels 1 and 3, and a second-order process (λ
(3)
1 ∼ 1/∆2) of three-

photon transitions between levels 1 and 4.
To conclude, we note that the method presented here is restricted to su(d) de-

formed algebras. In fact, many other phenomena can be modeled by Hamiltonians
quite similar to (4.8), namely

Ĥint = a Ŷ0 + g(Ŷ+ + Ŷ−) + Ĉ, (4.74)

where C is some integral of motion and a a constant. In these theories, the polynomial
deformation is defined in the following fashion in the Cartan-Weyl basis:

[Ŷ0, Ŷ±] = ±Ŷ±, [Ŷ−, Ŷ+] = Ψ(Ŷ0) = Φ(Ŷ0 + 1) − Φ(Ŷ0), (4.75)

where Φ(Ŷ0) are appropriate structure polynomials. A detailed study of the applica-
tions of these algebras for solving evolution problems in nonlinear quantum models
may be found in (Karassiov 1994). For these model, the machinery of small rota-
tions works well and constitutes the most systematic way of constructing effective
Hamiltonians.


5 Relative phase for two-mode fields

In a classical framework, in order to analyze optical fenomena related with interference
(such as, visibility of fringes) one needs to work with light beams with a well defined
phase between them. The polarization properties of light are directly related to this
type of analysis. In quantum optics it is essential to deal with new conditions which
allow us to provide a complete analysis of the interference phenomena from a quantum
point of view.

The outline of this chapter is as follows: in section 1 we derive a criterion for a two-
mode field to have a well-defined relative phase. After that, we review the definition of
the classical visibility of inteference fringes to continue taking a quantum-mechanical
view of visibility, called “generalized visibility”. In section 2, we study the polarization
structure and the invariance properties of quantum two mode fields. Finally, in section
3, we connect the idea of distance with the problem of assessing the polarization
characteristics of a quantum field, exploring a suitable definition of polarization degree
that avoids at least some of the diffculties that previous approaches based on Stokes
parameters encounter.

5.1 States with well-defined relative phase

5.1.1 Differential phase shifts

In this chapter we deal with two-mode fields, which will be denoted by subscripts H
and V , indicating horizontal and vertical components, respectively.

To derive a criterion for a well-defined relative phase, we start by considering
the action of differential phase shift; i.e., letting the two modes undergo free evolu-
tion for unequal times. The free evolution the system for a time τ is described by
Û(τ) = exp(−iĤ0τ), where the Hamiltonian is Ĥ0 = ω n̂, and ω and n̂ is the angular
frequency and number operator, respectively. Thus the evolution operator for two
modes undergoing free evolution for times τH and τV , respectively, becomes

Ĥ = e−iω(τH+τV )N̂/2 eiϕn̂12/2, N̂ = n̂H + n̂V , n̂12 = n̂H − n̂V , (5.1)


86 5 Relative phase for two-mode fields

where the differential equation phase shift is ϕ = ω(τH − τV ). Since the operator
exp[−iω(τH + τV )N̂/2] in (5.1) gives the same phase shift for all states with fixed
photon numbers, and (two-mode) states with different photon numbers are orthogonal
and cannot interfere, this operator can be neglected in the context of interference.
Hence the unitary differential phase shift operator is photon-number conserving and
can be written

ÛPS(ϕ) = exp(iϕ n̂12). (5.2)

We now turn our attention to what constitutes a state with a well-defined relative
phase. We take an operational approach and assign this property to any two-mode
state on which (at least) two different relative phases can be encoded and read out
with certainty. This definition avoids all complications with associating a well-defined
relative-phase with some relative phase operator, or with the properties of the relative
phase statistical distribution.

It is well known to encode either of two relative phases ϕH or ϕV so that they can
be read out with certainty, i.e., be projected onto orthogonal meter eigenstates, the
relation

〈ξ|ÛPS(ϕV − ϕH)|ξ〉 = 0, (5.3)

must be fulfilled. That is, some relative phase shift ϕ = ϕV − ϕH must render the
state UPS(ϕ)|ξ〉 orthogonal to |ξ〉. In our treatment, equation (5.3) constitutes thus
the mathematical criterion for a state with a well-defined relative phase.

In the following, the basis states of the excitation manifold N will be denoted as
|N, k〉 = |k〉H ⊗ |N − k〉V , (k = 0, 1, . . . , N) and the states |ξ〉 will be written as

|ξ〉 =

∞
∑

N=0

N
∑

k=0

CN,k |N, k〉. (5.4)

Since ÛPS(ϕ)|0, 0〉 = 0, it is clear that we must have 〈0, 0|ξ〉 = 0 in order to satisfy
equation (5.3). If one wishes to quantify the ability to distinguish between two relative
phases, one can define the distinguishability D (in the maximum-likelihood estima-
tion sense (Björk, Trifonov, Tsegaye and Söderholm 1998)) between the two relative
phases ϕH and ϕV by

D =

√

1 − 4pHpV |〈ξ|ÛPS(ϕH − ϕV )|ξ〉|2, (5.5)

where pH and pV are the a priori probabilities of encoding the phase shifts ϕH and
ϕV , respectively. This distinguishability limit is referred to as the Helstrom bound
(Helstrom 1976)and is well known in estimation theory. We see that for any nonzero
a priori probabilities pH and pV , unit distinguishability implies that equation (5.3)
has to be fulfilled.


5.1 States with well-defined relative phase 87

5.1.2 Classical visibility

Interference phenomena are directly related with the relative phase between modes.
From an experimental point of view, this gives interference fringes. Classically, the
visibility of these fringes give us an idea of the sharpeness of the relative phase. In
particular, if we cannot observe fringes, relative phase is random.

(a)                  Phase shift                 Detectors

E1(t)

             Beam splitter

E2(t)

(b)                  Phase shift                 Detectors

Іξ>

D1

D2

D1

D2

Generalized beam splitter Û

Fig. 5.1. (a) A schematic setup for a visibility experiment. (b) A quantum-mechanical
generalized visibility experiment.

We consider a Mach-Zehnder interferometer despicted in figure 5.1, to avoid any
unnecessary complication of more elaborated setups. Two fields interfere in a beam
splitter and the relative phase between them can be varied by a phase shifter. After
the beam splitter, the outgoing field intensities are monitored by two detectors. The
visibility is defined in terms of the ensemble-averaged modulation of the measured
intensities as a function of the relative-phase shift. To make a quantitative analysis
of classical visibility, we consider the interference of two harmonic waves (sufficiently
generally) described by

EH(t) = EH cos(ωt+ ϑ), EV (t) = EV cos(ωt), (5.6)


88 5 Relative phase for two-mode fields

where EH and EV are (real) electric field amplitudes. The phase shifter transforms
EH into

EH(t) = EH cos(ωt+ ϑ+ ϕ). (5.7)

The beam splitter is described by the unitary transformation

U(α) =

(

cosα sinα
− sinα cosα

)

, (5.8)

which assumes, without loss of generality, that reference planes have been chosen so
that all coefficients are real. If the beam splitter transmittance is neither zero nor
unity, the (time-averaged) intensity I detected by the two detectors is a sinousoidally
varying function of the phase shift ϕ.

The visibility for each detector is defined

V =
Imax − Imin

Imax + Imin
, (5.9)

where Imax and Imin are the maximum and minimum intensities measured by the
detector. In general, the visibility is not equal for the two detectors: to get unit
visibility one must arrange so that Imin = 0 while Imax 6= 0. Also note that the
intensity detected in each detector is a second-order correlation function. For the
special case when | cosα| = | sinα| = 1/

√
2, the detector signals provide a measure of

the complex degree of coherence between the two fields (Mandel and Wolf 1995).
Using equations (5.6), (5.8) and (5.9), it is straightforward to show that Imin = 0

for both detector, implies a phase shift ϕ fulfilling ϑ + ϕ = 0 or ϑ + ϕ = π. In the
first case, it is possible to get the intensity falling onto detector D1 to be zero if
tanα = −EH/EV . To make the intensity falling onto detector D2 be zero, one must
require tanα = EV /EH . If the phase ϕ is instead set to π − ϑ, then detector D1 sees
zero intensity for tanα = EH/EV and D2 sees zero intensity for tanα = −EV /EH . In
both cases, we can only get the visibility to equal unity in both beam splitter output
modes if EH = ±EV and | cosα| = | sinα| = 1/

√
2.

In summary, from the classical viewpoint, in order to be able to get unit visibility
in the sense just defined, the field amplitudes must be equal and the beam splitter
needs to have equal transmission and reflection and equal to 1/

√
2.

5.1.3 Quantum visibility

Let us now turn to the quantum mechanical situation depicted in figure 5.1b. One
mode of the state |ξ〉 is phase shifted by a relative amount ϕ to the other. Subse-
quently, the state is transformed by a generalized beam splitter, described by the
unitary transformation Û (whose properties will be defined below) and finally the
output modes of the generalized beam splitter are detected by two photon-counting


5.1 States with well-defined relative phase 89

detectors. The generalized visibility is, in analogy with the classical version, defined
in terms of the ensemble-averaged modulation of the detected photon number in the
respective photon detectors. The main difference between the quantum and the clas-
sical setups is that the generalized beam splitter is typically a nonlinear beam splitter
that has to match the impinging two-mode state. Different two-mode states require
different generalized beam splitters. If it is possible to find two relative phase shift
settings such that for one setting all the photons of state |ξ〉 impinge on detector D1,
and for the other setting all the photons impinge on detector D2, then we define the
state |ξ〉 to have unit generalized visibility. Comparing this definition with our oper-
ational definition of a state with a well-defined relative phase, one is lead to suspect
that the two definitions are interrelated. Let us next cast generalized visibility in a
mathematical framework. We begin by defining the quantum-mechanical visibility of
the setup in figure 5.1b in analogy with the classical visibility

V =
〈n̂〉max − 〈n̂〉min

〈n̂〉max + 〈n̂〉min
, (5.10)

where n̂ is the number operator associated with the pertinent detector. To get unit
generalized visibility in one of the output modes, we must have 〈n̂〉min = 0 while
〈n̂〉max 6= 0. This implies that for some suitable differential phase shift ϕ the vacuum
state |0〉 must impinge on the detector, since this is the only single-mode state with
〈n̂〉 = 0. Now consider a two-mode quantum state |ξ〉, which in general is entangled.
It is always possible to find a unitary and photon-number preserving transformation
Û such that

Û |ξ〉 = |ψ〉 ⊗ |0〉, (5.11)

i.e., the state |ξ〉 is transformed to a factorizable state with no excitation in one of the
modes. In a classical visibility measurement, Û is assumed to describe the action of a
50/50 beam splitter, but for a generalized visibility measurement we must only require
that Û is unitary and photon-number preserving. The latter requirement is reason-
able as we want interference, rather than photon loss, to determine the generalized
interference. A consequence of the photon-number conservation is that

Û |0, 0〉 = eiς |0, 0〉, (5.12)

where ς is a real number. In order to get unit generalized visibility in the measurement
depicted in the figure 5.1b it is necessary that if the state |ξ〉 is phase shifted by an
appropriate amount ϕ, Û must transform this new state to a state of the form |0〉⊗|ϕ〉.
Hence,

Û ÛPS|ξ〉 = |0〉 ⊗ |ϕ〉. (5.13)

Equations (5.10) and (5.12) together with the requirement that |ξ〉 6= |0, 0〉 are suffi-
cient and necessary conditions to get unit generalized visibility in the output modes
of a generalized beam splitter. To see what requirements these equations imply for


90 5 Relative phase for two-mode fields

the state |ξ〉, let us compute the scalar product between the final states (5.11) and
(5.13):

(〈ψ| ⊗ 〈0|)(|0〉 ⊗ |ϕ〉) = 〈ψ|0〉〈0|ϕ〉. (5.14)

Using the left hand sides of equations (5.11) and (5.12), the scalar product can be
rewritten as

〈ψ|0〉〈0|ϕ〉 = 〈ξ|0, 0〉〈0, 0|ξ〉 = 〈ξ|Û Û†ÛPS(ϕ)|ξ〉 = 〈ξ|ÛPS(ϕ)|ξ〉, (5.15)

where we have used the fact that vacuum is unaffected by a phase shift and equa-
tion (5.10) to arrive equation (5.14). A rearrangement now gives us our final result,
delineating a necessary and sufficient condition to achieve unit generalized visibility
as

〈ξ|ÛPS(ϕ)|ξ〉 − |〈ξ|0, 0〉|2 = 0. (5.16)

That the condition is sufficient follows from that any state fulfilling equation (5.16)
can also fulfill both equations (5.11) and (5.13), which is necessary and sufficient to
get unit generalized visibility. That the condition is necessary follows from the fact
that any state that does not fulfill the condition (5.16) can fulfill only one of equations
(5.10) and (5.12), not both.

We see that in order to get unit generalized visibility there must exist a differential
phase shift ϕ such that all the photon-number manifolds of the state ÛPS(ϕ)|ξ〉 are
rendered orthogonal to the manifolds of the initial |ξ〉. Comparing equations (5.3)
and (5.16), one sees that the former condition is stronger than the latter in that every
state fulfilling equation (5.3) will also fulfill equation (5.16), but a state satisfying
equation (5.16) does not necessarily satisfy equation (5.3). The physical reason is
that since visibility (both the classical and the generalized) is an ensemble-averaged
quantity, the fact that ÛPS(ϕ) does nothing to the state |0, 0〉 is not important to the
generalized visibility. Null photon counts will contribute neither to the minima nor to
the maxima (as ϕ is varied) of the photon interference pattern.

As long as the state |ξ〉 contains higher photon-number manifolds, and each of these
states are simultaneously rotated to an orthogonal state for some differential phase
shift ϕ, an interference pattern with unit generalized visibility can be observed. On
the contrary, in order to predict a phase shift with certainty (i.e., for every individual
detected state) the state must not contain any component of the vacuum state, since
every time the state ÛPS(ϕH) or ÛPS(ϕV ) collapses into the vacuum state it leads to
an inconclusive result of which phase shift ϕH or ϕV was used. Hence, the relation
(5.3) is sharper than the requirement of unit generalized visibility (5.16) in that every
state fulfilling equation (5.3) can also display unit generalized visibility, while a perfect
generalized visibility does not ensure that the relative phase of the state is precisely
defined.

Visibility and coherence are intimately connected. Let us therefore briefly discuss
the connection between generalized visibility and coherence. In terms of coherence


5.1 States with well-defined relative phase 91

theory, the detectors in figure 5.1b measure a superposition of even order correlation
functions. The explicit choice of Û will determine the particular superposition (the
fact that particle detectors are quadratic in the incident fields assures that no odd-
order coherence functions are measured). If the two-mode state is in a photon-number
eigenstate with a total of N photons, only the even-order correlation functions up to
set 2N are measured. This is due to the fact that the interaction Hamiltonian re-
alizing any particle number preserving, two-mode, unitary transformation Û can be
synthesized by a normally-ordered polynomial of order 2N in the creation and anni-
hilation operators of the fields (Björk, Söderholm and Karlsson 1998). The coherent
properties of a N -particle state is not simply given by the 2Nth order correlation
function (Mandel and Wolf 1995). Yet, the criterion for when a two-mode state has a
well-defined relative phase is surprisingly simple (5.3). A classical visibility measure-
ment is a special case of a generalized visibility measurement where Û has a particular
form (expressed in creation and annihilation operators it contains only the linear term
of each mode) so that no correlation functions higher than of the second order are
measured.

Let us show that unit generalized visibility does not require any symmetry of the
state |ξ〉 with the respect of permutation of modes. As demonstrated above this is
necessary in a classical experiment. To show this we construct a simple example, e.g.,
the state

|ξ〉 =

√

3

10
(|N, 0〉 + |N, 1〉) +

√

2

10
|N, 3〉, (5.17)

where N ≥ 3. This state has no symmetry with respect to the permutation of modes,
and its average excitation in the second mode is much larger than its excitation in
the first mode, if N is large. Yet, for ϕ = ± arctan(

√
15/4) ≃ ±0.42π rad, equation

(5.3) is satisfied. Hence, the state has a well-defined relative phase and can therefore
display unit generalized visibility.

Other interesting example, consists of the eigenstate of the relative-phase operator
(Luis and Sánchez-Soto 1993). The most general form of such an eigenstate in photon-
number manifold N can be written (Luis and Sánchez-Soto 1993)

|φ(N)
r 〉 =

1√
N + 1

N
∑

k=0

eikφ(N)
r |N, k〉, (5.18)

where φ
(N)
r = φ

(N)
0 + 2πr/(N + 1), (r = 1..N). Since the eigenstates are orthogonal,

they will fulfill equation (5.3) above for N 6= 0 and ϕV − ϕH = 2πkr/(N + 1). Thus,
these states have a well-defined relative phase. In spite of displaying nonunit classical
visibility, they can display unit generalized visibility. To give a specific example of
a unitary transformation which gives any state a unit generalized visibility, consider
the unitary transformation


92 5 Relative phase for two-mode fields

Û =

∞
∑

N=0

N
∑

r=0

|N, r〉〈ϕ(N)
r |. (5.19)

In every manifold, the state |ϕ(N)
r 〉 is transformed into the number difference state

|N, r〉.
We can conclude that generalized visiblility depends a lot of the generalized beam-

splitter choice. The most of the beam splitters do not give us unit visibility, in spite
of the state fulfills (5.3).

5.2 SU(2) invariance properties of two-mode fields

As we have seen above, classical visibility is not sufficient to know when a state is
well-defined relative phase. Furthermore, we have shown that in a new definition of
visibility (generalized visibility), we need to include higher-order correlations. In order
to get this, we have introduced generalized beam splitters as unitary transformations
preserving the total photon number. If we take this into account, it would be interest-
ing to analize what is the highest precission we can get in a relative-phase measure.

In 1852, Stokes devised a systematic treatment of two-mode states by introducing
the parameters that today bear his name. A great advantage with them is that they
are easily measured. To introduce these parameters, we assume a monochromatic
plane wave propagating in the z direction, whose electric field lies in the xy plane.

S0 = |EH |2 + |EV |2 , S1 = 2 Re(E∗
HEV ) ,

(5.20)

S2 = 2 Im(E∗
HEV ) , S3 = |EH |2 − |EV |2 ,

It is easy to show that S2
1 + S2

2 + S2
3 = 1. That is, the Stokes parameters of any

monochromatic, non-stochastic plane wave lie on the surface of the Poincaré sphere.
If the field amplitudes are fluctuating, then the degree of classical polarization is
defined as

Pcl =

√

S2
1 + S2

2 + S2
3

S0
. (5.21)

From a quantum point of view, a two-mode field that can be fully described by two
complex amplitude operators, denoted by âH and âV . The commutation relations of
these operators are standard:

[âj , â
†
k] = δjk , j, k ∈ {H,V } . (5.22)

The Stokes operators are then defined as the quantum counterparts of the classical
variables, namely (Alodjants and Arakelian 1999; Chirkin et al. 1993; Collett 1970;
Jauch and Rohrlich 1976)


5.2 SU(2) invariance properties of two-mode fields 93

Ŝ0 = â†H âH + â†V âV , Ŝ1 = â†H âV + â†V âH ,

(5.23)

Ŝ2 = i(âH â
†
V − â†H âV ) , Ŝ3 = â†H âH − â†V âV ,

and their mean values are precisely the Stokes parameters (〈Ŝ0〉, 〈Ŝ〉), where Ŝ =
(Ŝ1, Ŝ2, Ŝ3). Using the relation (5.22), one immediately gets that the Stokes operators
satisfy the commutation relations of angular momentum:

[Ŝ, Ŝ0] = 0 , [Ŝ1, Ŝ2] = 2iŜ3 , (5.24)

and cyclic permutations. The noncommutability of these operators precludes the
simultaneous exact measurement of their physical quantities. Among other conse-
quences, this implies that no field state (leaving aside the two-mode vacuum) can
have definite nonfluctuating values of all the Stokes operators simultaneously. This is
expressed by the uncertainty relation

(∆Ŝ)2 = (∆Ŝ1)
2 + (∆Ŝ2)

2 + (∆Ŝ3)
2 ≥ 2〈Ŝ0〉 . (5.25)

Contrary to what happens in classical optics, the electric vector of a monochromatic
quantum field never describes a definite ellipse (Luis 2002).

In mathematical terms, a linear polarization transformation is any transformation
generated by the operators Ŝ. It is well known (Yurke et al. 1986) that the oper-
ator Ŝ2 is the infinitesimal generator of geometrical rotations around the direction
of propagation, whereas Ŝ3 is the infinitesimal generator of differential phase shifts
between the modes. As indicated by equation (5.24), these two operators suffice to
generate all SU(2) transformations, which in experimental terms means that they can
be accomplished with a combination of phase plates and rotators (Björk et al. 2002).

The standard definition of the degree of polarization is now (Born and Wolf 1980;
Saastamoinen and Tervo 2004; Simon 1990)

Psc =

√

〈Ŝ〉2

〈Ŝ0〉
=

√

〈Ŝ1〉2 + 〈Ŝ2〉2 + 〈Ŝ3〉2

〈Ŝ0〉
, (5.26)

where the subscript sc indicates that this is a semiclassical definition, mimicking the
form of the classical in (5.21). In the semiclassical description it is implicitly assumed
that unpolarized light (i. e., the origin of the Poincaré sphere) is defined by the specific
values (Karassiov 1993)

〈Ŝ1〉 = 〈Ŝ2〉 = 〈Ŝ3〉 = 0 . (5.27)

Sometimes the extra requirement that the Stokes parameters are temporally invariant
is added to make the definition even more stringent (Barakat 1989). While this affords
a very intuitive image, it has also serious flaws that give rise to strange concepts such


94 5 Relative phase for two-mode fields

as that of quantum states with “hidden” polarization (Klyshko 1992). Actually, this
notion leads to the paradoxical conclusion that unpolarized light has a polarization
structure, which is latent when the mean intensities are measured and detectable
when the noise intensities are measured (Karassiov 1995). These paradoxes can be
traced back to the fact that the Stokes parameters are proportional to the second-
order correlations of the field amplitudes. This may be sufficient for most classical
problems, but for quantum fields higher-order correlations are crucial, as we saw in
the case of visibility. Hence, we need find a generalized polarization degree.

Today, there is a wide consensus (Björk et al. 2002; Wünsche 2003) in considering
unpolarized light as the only one described by quantum states that are invariant
with respect to any SU(2) polarization transformation, then there is no more any
“hidden” polarization. Any state satisfying this invariance condition will also fulfill
the classical definition of an unpolarized state, but the converse is not true. It has
been shown (Agarwal 1971; Lehner et al. 1996; Prakash and Chandra 1971) that the
density operator of such quantum unpolarized states can be always written as

σ̂ =
∞

⊕

N=0

λN 1̂1N , (5.28)

where N denotes the excitation manifold in which there are exactly N photons in
the field. All the coefficients λN are real and nonnegative and to meet the unit-trace
condition of the density operator they must satisfy

∞
∑

N=0

(N + 1)λN = 1 . (5.29)

5.3 Quantum degree of polarization as a distance

Measures of nonclassicality have been defined as the distance to an appropriate set
representing classical states (Dodonov et al. 2000; Hillery 1987; Marian et al. 2002).
Similarly, the minimum distance to the (convex) set of separable states has been
used to introduce measures of entanglement (Vedral et al. 1997). In the same vein,
we propose to quantify the degree of polarization as

P(ˆ̺) ∝ inf
σ̂∈U

D(ˆ̺, σ̂) , (5.30)

where U denotes the set of unpolarized states of the form (5.28) and D(ˆ̺, σ̂) is any
measure of distance (not necessarily a metric) between the density matrices ˆ̺ and σ̂,
such that P(ˆ̺) satisfies some requirements motivated by both physical and mathe-
matical concerns. The constant of proportionality in equation (5.30) must be chosen
in such a way that P is normalized to unity, i.e., sup ˆ̺ P(ˆ̺) = 1.


5.3 Quantum degree of polarization as a distance 95

In (Gilchrist et al. 2005) a check list of six simple, physically-motivated criteria
that should be satisfied by any good measure of distance between quantum processes
can be found. For our problem, we impose the following two conditions:

(C1) P(ˆ̺) = 0 iff ˆ̺ is unpolarized.
(C2) Energy-preserving unitary transformations ÛE leave P(ˆ̺) invariant; that is,

P(ˆ̺) = P(ÛE ˆ̺Û†
E).

The first condition is to some extent trivial: it ensures that unpolarized and only un-
polarized states have a zero degree of polarization. The second takes into account that
the requirement that an unpolarized state is invariant under any SU(2) polarization
transformation makes it also invariant under any energy-preserving unitary transfor-
mation (Sehat et al. 2005): these include not only the transformations generated by

Ŝ, but also those generated by Ŝ0, which, in technical terms, corresponds to the group
U(2) (Wünsche 2003).

It is clear that there are numerous nontrivial choices for D(ˆ̺, σ̂) (by nontrivial
we mean that the choice is not a simple scale transformation of any other distance).
None of them could be said to be more important a priori than any other, but the
significance of each candidate would have to be seen through physical assumptions.
To illustrate this point further, let us take an extreme example fulfilling the previous
conditions (Vedral et al. 1997). Define the discrete distance

Ddis(ˆ̺, σ̂) =







1 , ˆ̺ 6= σ̂ ,

0 , ˆ̺ = σ̂ .
(5.31)

If the degree of polarization is computed using this distance, we have

Pdis(ˆ̺) =







1 , ˆ̺ /∈ U ,

0 , ˆ̺ ∈ U .
(5.32)

This therefore tells us only if a given state ˆ̺ is unpolarized or not.
There are authors demanding that D(ˆ̺, σ̂) is a metric (Gilchrist et al. 2005). This

requires three additional properties:

1. Positiveness: D(ˆ̺, σ̂) ≥ 0 and D(ˆ̺, σ̂) = 0 iff ˆ̺ = σ̂.
2. Symmetry: D(ˆ̺, σ̂) = D(σ̂, ˆ̺).
3. Triangle inequality: D(ˆ̺, τ̂) ≤ D(ˆ̺, σ̂) +D(σ̂, τ̂).

These are quite reasonable properties, since most distances used in quantum mechan-
ics are based on an inner product and so they automatically fulfill them. However,
there exist pertinent examples in which D is not a metric. For example, the quantum
relative entropy (Donald 1986; Hiai and Petz 1991; Ohya 1989; Wehrl 1978)


96 5 Relative phase for two-mode fields

S(ˆ̺||σ̂) = Tr[ˆ̺(ln ˆ̺− ln σ̂)] (5.33)

is not symmetric and does not satisfy the triangle inequality. Nevertheless, it generates
a valuable measure of entanglement, and the corresponding degree of polarization
satisfies both C1 and C2.

For a detailed analysis we shall consider the Hilbert-Schmidt metric

DHS(ˆ̺, σ̂) = || ˆ̺− σ̂||2HS = Tr[(ˆ̺− σ̂)2] , (5.34)

which has been previously studied in the contexts of entanglement (Bertlmann et al.
2002; Ozawa 2000; Witte and Trucks 1999). Since DHS(ˆ̺, σ̂) is a metric, condition C1
is satisfied. It follows from the unitary invariance of the Hilbert-Schmidt metric that
also C2 is satisfied.

According to the general strategy outlined in (5.30), for a given state ˆ̺ we should
find the unpolarized state σ̂ that minimizes the distance

DHS(ˆ̺, σ̂) = Tr(ˆ̺2) + Tr(σ̂2) − 2Tr(ˆ̺σ̂) . (5.35)

If we take into account that the purity of an unpolarized state is

Tr(σ̂2) =
∞
∑

N=0

(N + 1) λ2
N , (5.36)

we easily get

DHS(ˆ̺, σ̂) = Tr(ˆ̺2) +

∞
∑

N=0

[(N + 1)λ2
N − 2pNλN ] , (5.37)

where pN is the probability distribution of the total number of photons

pN =

N
∑

k=0

̺Nk,Nk , (5.38)

and ̺Nk,N ′k′ = 〈N, k| ˆ̺|N ′, k′〉. Now, it is easy to obtain the coefficients λN that
minimize this distance. The calculation is direct and the result is

λN =
pN

N + 1
. (5.39)

The density operator σ̂opt ∈ U with these optimum coefficients λN satisfies the con-
straint (5.29) and hence minimizes the distance (5.35). Note that σ̂opt can be written
as

σ̂opt =

∞
∑

N=0

pN σ̂
(N)
opt , (5.40)


5.3 Quantum degree of polarization as a distance 97

with

σ̂
(N)
opt =

1

N + 1

N
∑

k=0

|N, k〉〈N, k| . (5.41)

With all this in mind, we can define the Hilbert-Schmidt degree of polarization by

PHS(ˆ̺) = Tr(ˆ̺2) −
∞
∑

N=0

p2
N

N + 1
, (5.42)

which is determined not only by the purity 0 < Tr(ˆ̺2) ≤ 1 (as it happens for other
measures (Heller 1987)), but also by the distribution of the number of photons pN .
Although the maximum Hilbert-Schmidt distance between two density operators is
2, the minimum distance to an unpolarized state is normalized to unity.

Using equations (5.36) and (5.39), the Hilbert-Schmidt degree of polarization can
be recast as

PHS(ˆ̺) = Tr(ˆ̺2) − Tr(σ̂2
opt) , (5.43)

which makes it easy to verify that it vanishes only for unpolarized states, in agreement
with the condition C1.

It has been shown (Bertlmann et al. 2002; Ozawa 2000; Witte and Trucks 1999)
that the Hilbert-Schmidt distance is not monotonically decreasing under every com-
pletely positive trace-preserving map (what is called the CP nonexpansive property).
This has motivated that the quantum information community has identified the fi-
delity as a particularly important alternative approach to the definition of a distance
measure for states (Nielsen and Chuang 2000).

In consequence, as our second candidate of distance we will employ the fidelity (or
Uhlmann transition probability) (Uhlmann 1976)

F (ˆ̺, σ̂) = [Tr(σ̂1/2 ˆ̺ σ̂1/2)1/2]2 . (5.44)

A word of caution is necessary here. There is an ambiguity in the literature: both the
quantity (5.44) and its square root have been referred to as the fidelity. The reader
should take this into account when comparing different sources.

The fidelity has many attractive properties. First, it is symmetric in its arguments
F (ˆ̺, σ̂) = F (σ̂, ˆ̺), a fact that is not obvious from equation (5.44), but which follows
from other equivalent expressions. It can also be shown that 0 ≤ F (ˆ̺, σ̂) ≤ 1, with
equality in the second inequality iff ˆ̺ = σ̂. This means that the fidelity is not a
metric as such, but serves rather as a generalized measure of the overlap between
two quantum states. A common way of turning it into a metric is through the Bures
metric

DB(ˆ̺, σ̂) = 2[1 −
√

F (ˆ̺, σ̂)] . (5.45)


98 5 Relative phase for two-mode fields

The origin of this distance can be seen intuitively by considering the case when ˆ̺ and
σ̂ are both pure states. The Bures metric is just the Euclidean distance between the
two pure states, with respect to the usual norm on the state space.

Since the larger the fidelity F (ˆ̺, σ̂), the smaller the Bures distance DB(ˆ̺, σ̂), we
can define the Bures degree of polarization as

PB(ˆ̺) = 1 − sup
σ̂∈U

√

F (ˆ̺, σ̂) . (5.46)

An alternative definition would be 1 − supσ̂∈U F (ˆ̺, σ̂), which arises naturally in the
context of quantum computation (Gilchrist et al. 2005). These definitions order the
states ˆ̺ in the same way. Unfortunately, we have not found a general expression of the
unpolarized state σ̂ that gives the maximum fidelity. Such a task must be performed
case by case and will be illustrated with some selected examples.

5.3.1 Some examples

From equation (5.42) we infer that all pure N -photon states have the same Hilbert-
Schmidt degree of polarization. For such states, we have

P
(N)
HS =

N

N + 1
. (5.47)

The Bures degree of polarization for these states can also be readily found:

P
(N)
B = 1 − 1√

N + 1
. (5.48)

The vacuum is the only unpolarized state, in agreement with condition C1. Note also
that the expressions (5.47) and (5.48) apply, e. g., to the states |n〉H ⊗ |n〉V . Since

for them 〈Ŝ〉 = 0, classically they would be unpolarized for every n (that is, Psc = 0,
even in the limit n ≫ 1). In our distance-based approach, the degree of polarization
is a function of all moments of the Stokes operators and not only of the first one, as it
happens for Psc, which causes this quite different behavior. We also observe that all
these states lying in the (N + 1)-dimensional invariant subspace satisfy P → 1 when
their intensity is increased.

Next, we define the diagonal states as those that can be expressed as

ˆ̺diag =
∞
∑

N=0

N
∑

k=0

pNk|Ψ (N)
k 〉〈Ψ (N)

k | , (5.49)

where we let pNk ≥ pNk+1, for all k < N , and {|Ψ (N)
k 〉}N

k=0 is an arbitrary orthonormal
basis in the excitation manifold N . It then follows from C2 that any two diagonal


5.3 Quantum degree of polarization as a distance 99

states whose probability distribution {pNk}N
k=0 coincide, must have the same degree

of polarization. For any diagonal state, we have

PHS(ˆ̺diag) =

∞
∑

N=0

N
∑

k=0

p2
Nk −

∞
∑

N=0

p2
N

N + 1
≤

∞
∑

N=0

Np2
N

N + 1
. (5.50)

To deal with the Bures degree of polarization for this example, we first note that

√

F (ˆ̺diag, σ̂) =
∞
∑

N=0

N
∑

k=0

√

λN pNk =
∞
∑

N=0

sN

√

λN , (5.51)

where

sN =

N
∑

k=0

√
pNk . (5.52)

The extremal points of (5.51) are then determined by

sN

2
√
λN

− µ(N + 1) = 0 , (5.53)

where µ is a Lagrange multiplier that takes into account the constraint (5.29). Solv-
ing for λN and imposing again equation (5.29) to fix the value of µ, the optimum
parameters λN are found to be

λN =
s2N

(N + 1)2
∞
∑

k=0

s2k
k + 1

. (5.54)

In this way, we finally arrive at

PB(ˆ̺diag) = 1 −

√

√

√

√

∞
∑

N=0

s2N
N + 1

. (5.55)

One can easily prove that

√
pN ≤ sN ≤

√

(N + 1)pN , (5.56)

so we have the bound

PB(ˆ̺diag) ≤ 1 −

√

√

√

√

∞
∑

N=0

pN

N + 1
< 1 . (5.57)


100 5 Relative phase for two-mode fields

This and equation (5.42) show that
∑∞

N=0 p
2
N (N + 1)−1 → 0 is necessary for both

PB(ˆ̺diag) and PHS(ˆ̺) to approach unity. The latter also requires the purity to ap-
proach unity, whereas it is clear from equation (5.57) that this is not necessary in
order to have PB(ˆ̺) → 1.

As another relevant example, let us consider the case in which both modes are in
(quadrature) coherent states. The product of two quadrature coherent states, which
we shall denote by |αH , αV 〉, can be expressed as a Poissonian superposition of SU(2)
coherent states (Atkins and Dobson 1971)

|αH , αV 〉 =

∞
∑

N=0

e−N̄/2 N̄
N/2

√
N !

|N,ϑ, ϕ〉 , (5.58)

where N̄ = |αH |2 + |αV |2 is the average number of excitations and the SU(2) coherent
states are defined as (Perelomov 1986)

|N,ϑ, ϕ〉 =

N
∑

k=0

(

N
k

)1/2 (

sin
ϑ

2

)N−k (

cos
ϑ

2

)k

e−ikϕ |N, k〉 , (5.59)

and the state parameters are connected by the relations

αH = e−iϕ/2
√

N̄ sin
ϑ

2
, αV = eiϕ/2

√

N̄ cos
ϑ

2
. (5.60)

Taking into account that

∞
∑

N=0

p2
N

N + 1
=

I1(2N̄)

N̄
e−2N̄ , (5.61)

where I1(z) is the modified Bessel function, equation (5.42) reduces to

PHS = 1 − I1(2N̄)

N̄
e−2N̄ . (5.62)

When N̄ ≫ 1 we can retain the first term in the asymptotic expansion of I1(z) to
obtain

PHS ≃ 1 − 1

2
√
πN̄3/2

. (5.63)

As a last example, we consider the maximally entangled states

|ζ〉 = Ŝ(ζ) (|0〉H ⊗ |0〉V ), (5.64)

where Ŝ(ζ) is the two-mode squeezing operator


5.3 Quantum degree of polarization as a distance 101

Ŝ(ζ) = exp(ζ∗âH âV − ζâ†H â
†
V ). (5.65)

These states are very important because are generated in typical parametric amplifi-
cation processes.

Using the disentangled form of Ŝ(ζ), the state (5.64) can be represented as

|ζ〉 =
1

cosh ζ

∞
∑

n=0

(−1)n tanhn ζ |n〉H ⊗ |n〉V , (5.66)

where we assume that ζ is real. In terms of the SU(2) invariant subspaces, (5.66)
reads as

|ζ〉 =
1

cosh ζ

∞
∑

N=0

(−1)N/2(tanh ζ)N/2 |N,N/2〉. (5.67)

This means that the total photon number distribution of this state is

pN =















(tanh ζ)N

cosh2 ζ
, for N even,

0, for N odd.

(5.68)

and its average number of excitations can be expressed as N̄ = 2 sinh2 ζ.
Using this form of pN and after some calculations, we get

∞
∑

N=0

p2
N

N + 1
=

1

2 cosh4 ζ tanh2 ζ
ln

(

1 + tanh2 ζ

1 − tanh2 ζ

)

, (5.69)

and hence the Hilbert-Schmidt degree of polarization is

PHS(|ζ〉) = 1 − 4ζ

sinh2(2ζ)
. (5.70)

When N̄ ≫ 1, we can approximate (5.70) by

PHS(|ζ〉) ≃ 1 − 4ζ exp(−4ζ), (5.71)

so, the degree of polarization tends to unity exponentially.
If we express (5.71) in terms of N̄ the polarization degree take the form

PHS(|ζ〉) ≃ 1 − 2 ln(N̄/2)

N̄2
. (5.72)

To deal with the Bures degree of polarization. We have to calculate the fidelity
between the density matrix for the state (5.64) and a generic σ̂. One immediately gets
that


102 5 Relative phase for two-mode fields

σ̂1/2 ˆ̺σ̂1/2 = (1 − |η|2)
∞
∑

N,N ′=0

ηN/2
√
λN√

N + 1

η∗N ′/2
√
λN ′√

N ′ + 1
|N,N/2〉〈N ′, N ′/2|. (5.73)

with η = tanh(ζ). Note that, given the particular form of these states, we have

(σ̂1/2 ˆ̺σ̂1/2)1/2 =
√

(1 − |η|2)
∞
∑

N,N ′=0

ηN/2 √
λN√

N+1

η∗N′/2
√

λN′√
N ′+1

×|N,N/2〉〈N ′, N ′/2|. (5.74)

In consequence
√

F (ˆ̺, σ̂) =
√

(1 − |η|2)
∞
∑

N=0

|η|NλN

N + 1
(5.75)

and the minimum of this fidelity is attained when λN = δN,0. In consequence, the
Bures polarization degree take the form

PB(ˆ̺) = 1 − 1

cosh ζ
, (5.76)

which tends again exponentially to the unity.
One can ask if the Hilbert-Schmidt and Bures measures order some pairs of states

differently. In the Appendix B we show that this is indeed the case, and the induced
degrees of polarization are therefore fundamentally different.

5.3.2 Maximally polarized states

A number of key concepts in quantum optics can be concisely quantified in terms
of distance measures. The notions of nonclassicality and entanglement to cite only a
few relevant examples, have been systematically formulated within this framework.
A good deal of effort has been devoted to characterize maximally nonclassical or
entangled states. However, maximally polarized states have been not considered thus
far, except for some trivial cases. It is precisely the purpose of this section provide a
complete description of such states, as well as feasible experimental schemes for their
generation.

To simplify as much as possible the discussion, we restrict our considerations here
to the Hilbert-Schmidt degree of polarization and drop the corresponding subscript.
It is clear from equation (5.42) that for the states living in the manifold with exactly
N photons, the optimum is reached for pure states. In fact, all such pure states have
the same degree of polarization (5.48) showing a typical scaling N−1, when N ≫ 1.
In particular, the important SU(2) coherent states are an example of these states.

However, this scaling law N−1 can be easily surpassed. Perhaps the simplest ex-
ample is when both modes are in (quadrature) coherent states |αH , αV 〉 [described


5.3 Quantum degree of polarization as a distance 103

from (5.58) to (5.60)], the polarization degree of these states (5.64) has a scaling
N−3/2. Other important example is when the state is a maximally entangled of the
form (5.64). In this case, the polarization has a scaling N−2.

In consequence, we are led to find optimum states for a fixed average number
of photons N̄ . Obtaining the whole optimum distribution pN in (5.42) is exceedingly
difficult, since it involves optimizing over an infinite number of variables. Our strategy
to attack this problem is to truncate the Hilbert space and consider only photon
numbers up to some value D, where we take the limit D → ∞ at the end. In this
truncated space, we need to find the states that maximize (5.42) with the constraints

pN ≥ 0,

D
∑

N=0

pN = 1,

D
∑

N=0

N pN = N̄ . (5.77)

It is clear that the optimum must be again pure states. If we introduce the notations
pT = (p0, p1, . . . , pL) and H = 2 diag[1, 1/2, . . . , 1/(D + 1)], the task can be thus
recast as

minimize
1

2
pTHp

(5.78)

subject to Ap = b,

p ≥ 0,

where

b =

(

1
N̄

)

, A =

(

1 1 1 · · · 1
0 1 2 · · · D

)

. (5.79)

We deal then with a quadratic program that, in addition, is convex, because H is
positive definite (Boyd and Vandenberghe 2004). The optimum point exists and it is
unique: in fact, there are numerous algorithms that compute this optimum in a quite
efficient manner. Alternatively, we may try to determine it analytically by incorpo-
rating the constraints by the method of Lagrange multipliers. The functional to be
minimized is

L(p, λ) =
1

2
pTHp− λT (Ap− b) . (5.80)

The first-order optimality conditions ∇L(p, λ) = 0 together with the initial equality
constraint, give the system of linear equations

(

H −AT

A 0

) (

p
λ

)

=

(

0
b

)

, (5.81)

whose formal solution is

λ = (AH−1AT )−1b , p = H−1ATλ . (5.82)


104 5 Relative phase for two-mode fields

Fig. 5.2. Optimum distribution pN , obtained by solving numerically the quadratic program
(5.78), plotted as a function of the average number of photons N̄ and N . We have taken N̄
running from 0.2 to 1 and the dimension of the space D = 4.

Before working out the analytical form of (5.82), in figure 5.2 we have plotted the
numerical solution of the quadratic program (5.78) for some values in 0 ≤ N̄ ≤ 1,
using the MINQ code implemented in Matlab. The number of nonzero components of
pN is [2N̄+1], where the brackets denote integer part. The distribution presents a clear
skewness and one can check that it can be well fitted to a Poisson distribution, which
in physical terms means that, in this range, a quadrature coherent state |αH , αV 〉
can be considered as optimum. To better assess this behavior, we have calculated the
associated Mandel Q parameter (Mandel and Wolf 1995)

Q =
〈(∆N̂)2〉

〈N̂〉
− 1 , (5.83)

where 〈(∆N̂)2〉 is the variance, which is a standard measure of the deviation from the
Poisson statistics. In figure 5.3 we have represented Q in terms if N̄ . As we can see,
Q increases linearly with N̄ and is zero only near N̄ ≃ 3.

In figure 5.4 we have plotted the optimum distribution pN for different integer
values of N̄ running from 1 to 9. The truncation value has been chosen to be 25 in
all the cases, although it is sufficient to ensure, for each value of N̄ , that pN ≃ 0 for
N > D. Three distinctive features can be immediately discerned: the solutions are


5.3 Quantum degree of polarization as a distance 105

Fig. 5.3. Plot of the Mandel Q parameter for the optimum distribution pN obtained nu-
merically from (5.78) in terms of the average number of photons N̄ of the state.

symmetric around N̄ , they are parabolic, and extend in a range from 0 to 2N̄ . The
two first facts are in agreement with the symmetry properties of the original problem
(5.78). The third one means a variance that scales as N̄2, at difference of what happens
for standard coherent optical processes presenting a variance linear with N̄ (as for,
e.g., in Poissonian or Gaussian statistics). In other words, the optimum states are
extremely noisy and fluctuating. When N̄ is not integer (or semi-integer), one can
appreciate a small asymmetry that is less and less noticeable as N̄ increases.

We conclude that we can take the dimension D to be 2N̄ without serious error.
Given the very simple form of H and A, we can express the final solution (5.82) in a
closed analytic form:

pN = 3
(N̄ + 1)2 − (N − N̄)2

(2N̄ + 1)(N̄ + 1)(2N̄ + 3)
≃ 3

2N̄

(

N

N̄
− 1

2

N2

N̄2

)

, (5.84)

which is properly normalized and shows all the aforementioned characteristics, with
a maximum value of pN̄ ≃ 3/(4N̄). If we use x = N/(2N̄), which can be taken
as a quasicontinuous variable 0 ≤ x ≤ 1, we can convert (5.84) in equation p(x) =
(3/N̄) x(1−x), which is the Beta distribution of parameters (2, 2) (Evans et al. 2000).
For the solution (5.84), the corresponding degree of polarization is

Popt = 1 − 3

(2N̄ + 1)(2N̄ + 3)
∼ 1 − 3

4N̄2
. (5.85)

This provides a full characterization of the optimum states we were looking for. How-
ever, their physical implementation stands as a serious problem. The crucial issue for


106 5 Relative phase for two-mode fields

0
0

0.2

0.4

10
15

25
20

5

1
3

89

56
7

4
2

N
P

N
N

Fig. 5.4. Optimum distribution pN plotted as a function of the average number of photons
N̄ and N . We have taken N̄ to be a integer running from 1 to 9 and the dimension of the
space D = 25.

the scaling in (5.85) is the fact that distribution variance is proportional to N̄2. It
turns out that, for the discrete uniparametric distributions usually encountered in
physics, this is distinctive of the thermal (or geometric) distribution

pN =
1

N̄ + 1

(

N̄

N̄ + 1

)N

. (5.86)

But this is the photon statistics associated with the states (5.66) which are precisely
the twin beams generated in an optical parametric amplifier with a vacuum-state
input. The distribution (5.86) presents a skewness absent in the exact solution (5.84),
but a calculation of the state degree of polarization gives

P ≃ 1 − 2 ln(N̄/2)

N̄2
. (5.87)

Apparently, this is different from (5.85), but as soon as N̄ ≫ 1 they both ap-
proach unity in essentially the same way, which means that the (maximally entangled)
squeezed vacuum (5.66) is very close to optimum when N̄ ≫ 1.

Before ending, two important remarks seem in order. First, we observe that in
classical optics fully polarized fields have a perfectly defined relative phase between
H- and V -polarized modes (Brosseau 1998). Such a relation does not necessarily hold


5.3 Quantum degree of polarization as a distance 107

in the quantum domain: while the quadrature coherent states (5.59) have a sharp
relative phase, the twin-photon beams (5.66) have an almost random relative phase.
Second, the maximally polarized states we have found have a highly nonclassical
behavior, even in the limit N̄ ≫ 1, which makes the classical limit of these polarized
states a touchy business.


6 Conclusions

In this work, we have studied discrete quantum systems. In the following, we summa-
rize the most important results that have been obtained:

– We have analysed the basic properties of two- and three-dimensional quantum sys-
tems. We have shown their simmetries and how one can describe these systems. To
this end, we have studied and compared several possible descriptions of the phase
based in a polar decomposition, in a POVM or as the complementary variable to
amplitude.

– We have investigated an appropriate operator for the description of the relative
phase in the interaction of two-level and three-level atoms with quantum fields.
We have resorted to a proper polar decomposition of the corresponding ampli-
tudes, which has been justified on physical grounds as well as using the theory of
polynomial deformations of su(2) and su(3), respectively. The finite dimension of
the invariant subspaces implies that the spectrum of the relative phase is discrete,
which is very surprising from a physical point of view. From the phase states ob-
tained in this procedure, we have defined a probability distribution function for
the relative phase and studied its time evolution, showing how the formalism could
be applied to understanding more involved phenomena.

– Inspired by the behavior of linear models with simmetry su(2), we have introduced
small “rotations” generated by polinomial deformations. When a controlable pa-
rameter (normally the inverse of a detuning) becomes small, our method allows
us to describe the original model in terms of an effective Hamiltonian.

– We have shown a criterion to define a two-mode state with well-defined relative
phase. The relative-phase eigenstates satisfy this criterion in spite of displaying
less than unity visibility. This has led us to define a generalization of a visibility
measure, and subsequently to derive a criterion for when the generalized visibility
can be equal to unity. We have shown that all states with a well-defined relative
phase can display unit generalized visibility, whereas the converse is not true.

– Quantum optics entails polarization states that cannot be suitably described by
the (semi)classical formalism based on Stokes parameters. We have advocated the
use of a degree of polarization based on an appropriate distance to the set of


110 6 Conclusions

unpolarized states. Such a definition is closely related to other recent proposals
in different areas of quantum optics and is well behaved even when the classical
formalism fails. Finally, taking into account this, we have provided a complete
description of maximally polarized states, as well as feasible experimental schemes
for their generation.


A Positive operator-valued measures

In quantum mechanics it is fundamental to understand the form in which we extract
the information from the physics systems. This form is very important because the
taking of measures entails a projection of the system over one of the eigenstates of
the quantum variable that this being measured. The successive measurements on
equivalent systems provide us the statistic of the observable for each one of their
possible results.

Given a system, the state at the initial moment (this is before making some mea-
surement) can be represented, in general, like a linear superposition of the eigenstates
of an observable which we desire to measure

To be more precise, it turns out suitable to observe that, given an obsevable M̂ ,
the state of the system |ψ〉 can be expressed as

|ψ〉 =
∑

a

αa|ua〉, (A.1)

where |ua〉 are the ortogonal eigenstates of the operator M̂

M̂ =
∑

a

|ua〉Ma〈ua|, (A.2)

and Ma are the eigenvalues of M̂ .
We will detect a value Ma for the observable M̂ with probability |αa|2, in in that

case we will have prepared the system in an eigenstate of M̂ of |ua〉. One can conclude,
that the initial state |ψ〉 or quantum system, is projected on |ua〉 with probability
|〈ua|ψ〉|2. This forms the model of orthogonal measurement of Von Neumann.

A.1 Generalized measurements

If one try to generalize the measurement concept beyond these orthogonal measure-
ments considered by Von Neummann, one way to arrive at that idea is supose that
the sistem A is extended from the Hilbert state HA to a tensor product HA ⊗ HB ,


112 A Positive operator-valued measures

and that we perform orthogonal measurements in the tensor product, which will not
necessarily be orthogonal measurements in A.

We assume that the Hilbert space HA is a part of a larger space that has structure
of a direct sum

H = HA ⊕HB . (A.3)

The observers living in HA have acces only to observables with support in HA, ob-
servables M̂A such that

M̂A|ψ⊥〉 = 0 = 〈ψ⊥|M̂A, (A.4)

for any |ψ⊥〉 ∈ H⊥
A . Anyway, when we perform orthogonal measurements in H, prepar-

ing one of a set of mutually orthogonal states (eigenstates of a observer M), the
observer will know only about the component of that state in his space HA. Since
these components are not necessarily orthogonal in HA, he will conclude that the
measurement prepares one of a set or non-orthogonal states.

Suppose that the initial density matrix ˆ̺A has support in HA, (this is reasonable
because the observer will build the eigenstate in his enviroment) and that we perform
an orthogonal measurement in H. We will consider the case in which each Ea, is one-
dimensional projector, which will be general for our purposes. Thus , Êa = |ua〉〈ua|,
where |ua〉 is a normalized vector in H. This vector has a unique orthogonal decom-
position

|ua〉 = |ψ̃a〉 + |ψ̃⊥
a 〉, (A.5)

where |ψ̃a〉 and |ψ̃⊥
a 〉 are (unnormalized) vectors in HA and H⊥

A respectively. After the
measurement, the new density matrix will be |ua〉〈ua| with probability 〈ua| ˆ̺A|ua〉 =
〈ψa| ˆ̺A|ψa〉 since ˆ̺A has no support in H⊥

A .
But to the observer who knows nothing of HA, there is no physical distinction

between |ua〉 and |ψ̃a〉 (aside from normalization). If we write |ψ̃a〉 =
√
λa|ψa〉, where

|ψa〉 is a normalized state, then for the observer limited to observations in HA, we
might as well say that the outcome of the measurement is |ψa〉〈ψa| with probability
〈ψa| ˆ̺A|ψa〉.

Let us define an operator

∆̂a = ÊAÊaÊA = |ψ̃a〉〈ψ̃a| = λa|ψa〉〈ψa|, (A.6)

where ÊA is the orthogonal projection taking H to HA. Then we may say that the
outcome a has a probability P (a) = Tr(∆̂a̺). It is evident that each ∆̂a is hermitian
and nonnegative, but these are not projections unless λa = 1. Furthermore, ∆̂a fullfills

∑

a

∆̂a = ÊA

∑

a

(Êa)ÊA = ÊA = 1̂1A, (A.7)

that is, the sum of ∆̂a gives the identity in A.


A.2 Example: POVMs for angular variables 113

A partition of unity by nonnegative operators ∆̂a is called a positive operator-
valued measure (POVM). In summary, the set operators ∆̂a must fulfill

∆̂†
a = ∆̂a, ∆̂a ≥ 0,

∑

a

(∆̂a) = 11, (A.8)

In general, each outcome has a probability that can be expressed as

P (a) = Tr(∆̂a̺), (A.9)

hence, the second and third conditions of (A.8) guarantee that the probabilities are
nonnegative and with sum the unity respectively.

In the analized situation, we have imposed that the operators are one-dimensional.
The generalation of this is trivial. The main advantaje of we use POVMs is that we
do not need a description of the observables in terms of hermitian operators to get a
complete description of these observables. Thus, we can obtain the probability distri-
bution imposing additional restrictions to the restrictions imposed in (A.8), based on
fundamental properties of the observables that we want to describe.

A.2 Example: POVMs for angular variables

Se trata de obtener, a través del formalismo de POVM explicado anteriormente, las
POVMs que describen variables ángulo de manera general. Para ello, analizaremos las
restricciones adicionales que son necesarias para la contrucción de dicho formalismo.

When dealing with generic angle-action variables, one imposes that the complex
exponential of the angle (denoted by Ê) and the action variable (denoted by L̂) satisfy

[Ê, L̂] = Ê. (A.10)

If Ê were unitary, its action on the basis of eigenstates of L̂ (denoted by |m〉) will be
as a ladder operator

Ê|m〉 = |m− 1〉. (A.11)

The eigenstates of Ê (denoted by |θ〉) provide then an adequate description of the
quantum angle (Luis and Sánchez-Soto 1998). To ensure that ∆̂(θ) provides a mean-
ingful description of the angle as a canonically conjugate variable with respect L we
require

eiθ′L̂ ∆̂(θ) e−iθ′L̂ = ∆̂(θ + θ′), (A.12)

which reflects nothing but the basic feature that an angle shifter is an angle-
distribution shifter. This condition restricts the form of the POVM to

∆̂(θ) =
1

2π

∞
∑

n,m=0

bn,m ei(m−n)θ |m〉〈n|. (A.13)


114 A Positive operator-valued measures

We must take also into account that a shift in L̂ should not change the angle distri-
bution. A shift in L̂ is expressed by the operator Ê, since it shifts the distribution of
L̂ by one step. Therefore, we require as well

Ê ∆̂(θ) Ê† = ∆̂(θ), (A.14)

which, loosely speaking, is the physical translation of the fact that angle should be
complementary to the action variable. This implies the invariance

bn,m = bn−m, (A.15)

that allows us to recast Eq. (A.13) as

∆̂(θ) =
1

2π

∞
∑

ν

b−νe
−iνθÊν , (A.16)

while conditions (A.8) read now as now

|bν | ≤ 1, b∗ν = bν . (A.17)

Expressing Ê in terms of its eigenvectors |θ〉, we finally arrive at the general form of
a POVM describing the angle variable and fulfilling the natural requirements (A.12)
and (A.14):

∆̂(θ) =

∫

2π

dθ′ B(θ′) |θ + θ′〉〈θ + θ′|, (A.18)

where

B(θ) =
1

2π

∞
∑

ν=0

bνe
iνθ. (A.19)

This convolution shows that this effectively represents a noisy measurement, the func-
tion B(θ) giving the resolution provided by this POVM (Luis and Sánchez-Soto 1998).


B Hilbert-Schmidt and Bures degrees of
polarization

In this appendix, we will show that the Hilbert-Schmidt and the Bures distances
induce fundamentally different degrees of polarization. To this end, we consider the
states

ρ̂N1N2
=

2
∑

j,k=1

ρjk|Ψ (Nj)〉〈Ψ (Nk)| , (B.1)

where |Ψ (N1)〉 and |Ψ (N2)〉 are orthogonal pure states with N1 and N2 photons, re-
spectively. We here assume that N1 6= N2, and note that ρ̂NN is a diagonal state of
the form (5.49). To simplify calculations, we shall use the notation

p = ρ11 , 1 − p = ρ22 , q = ρ12 = ρ∗21 . (B.2)

The states |Ψ (N1)〉 and |Ψ (N2) then correspond to p = 0 and p = 1, respectively, and
the purity becomes

Tr(ρ̂2
N1N2

) = p2 + (1 − p)2 + 2|q|2 . (B.3)

We note in passing that 1 − 2p(1 − p) ≤ Tr(ρ̂2
N1N2

) ≤ 1 and 0 ≤ |q|2 ≤ p(1 − p).

In the basis (|Ψ (N1)〉, |Ψ (N2)〉), we can write

σ̂1/2ρ̂σ̂1/2 =





λN1
p

√

λN1
λN2

q

√

λN1
λN2

q∗ λN2
(1 − p)



 . (B.4)

Since the eigenvalues of this matrix are

χ± =
1

2
{λN1

p+ λN2
(1 − p) ±

√

[λN1
p− λN2

(1 − p)]2 + 4λN1
λN2

|q|2
}

, (B.5)

the fidelity can be expressed as

F (ρ̂, σ̂) = χ+ + 2
√
χ+χ− + χ− = λN1

p+ λN2
(1 − p)

+ 2
√

λN1
λN2

[p(1 − p) − |q|2] . (B.6)


116 B Hilbert-Schmidt and Bures degrees of polarization

For any fixed λN1
, λN2

, and p, the fidelity decreases as |q|2 increases. This could be
expected, since the unpolarized states do not have any off-diagonal elements.

The restriction (5.29) implies for this problem that

λN2
=

1 − (N1 + 1)λN1

N2 + 1
. (B.7)

In consequence, the coefficients that optimize the fidelity are determined by

∂F

∂λN1

= 0 = p− (1 +N1)(1 − p)

1 +N2

+ [1 − 2λN1
(1 +N1)]

√

p(1 − p) − |q|2
λN1

[1 − λN1
(1 +N1)](1 +N2)

.

(B.8)

We first consider pure states, for which |q|2 = p(1 − p). Choosing λN1
according

to

λN1
= 0 , p <

1 +N1

2 +N1 +N2
,

0 ≤ λN1
≤ 1

1 +N1
, p =

1 +N1

2 +N1 +N2
,

λN1
=

1

1 +N1
, p >

1 +N1

2 +N1 +N2
,

(B.9)

then maximizes the fidelity:

sup
σ̂∈U

F (ρ̂, σ̂) =



















1 − p

1 +N2
, p ≤ 1 +N1

2 +N1 +N2
,

p

1 +N1
, p ≥ 1 +N1

2 +N1 +N2
.

(B.10)

On the other hand, when |q|2 6= p(1 − p) (i.e., when 0 < p < 1), the solution of
equation (B.8) is

λN1
=

1

2(N1 + 1)

[

1 − (1 +N1)(1 − p) − (1 +N2)p
√

[1 +N1(1 − p) +N2p]2 − 4(1 +N1)(1 +N2)|q|2

]

. (B.11)

Depending on the parameters, this solution can take any value in the interval 0 <
λN1

< 1/(1 + N1). In fact, one can check that the choice (B.11) gives the closest
unpolarized state. Combining equations (B.6), (B.7), and (B.11), thus allows one to
obtain the fidelity and hence PB.


B Hilbert-Schmidt and Bures degrees of polarization 117

0.3

0.6

0.7

0.8

0.5

0.2 1.00.80.60.40

P H
S

Probability p

C
|q|2 = p (1 - p)

|q|2 = 0
B

0.25

0.40

0.45

0.50

0.35

0.2 1.00.80.60.40

P B A

|q|2 = p (1 - p)

|q|2 = 0

B

Probability p

0.4

0.2

0.55

0.30

Fig. B.1. Polarization degrees for two-dimensional states with N1 = 1 and N2 = 2. For any
given p, the maximum and minimum fidelities are given by |q|2 = p(1 − p) and |q|2 = 0,
respectively. Region A corresponds to states satisfying p > 4/7 and PB < 1 −

√

3/7. For
any state ρ̂A in this region,we have PB(ρ̂A) < PB(ρ̂B) while PHS(ρ̂A) > PHS(ρ̂B), where
the state ρ̂B is characterized by p = 4/7 and |q|2 = 0. In region C, the states satisfy
p > 2/3 and PHS > 2/3. For any such state ρ̂C , we have PHS(ρ̂C) > PHS(|Ψ (2)〉) and
PHS(ρ̂C) < PHS(|Ψ (2)〉), where |Ψ (2)〉 is the (arbitrary) pure two-photon state corresponding
to p = 0.


118 B Hilbert-Schmidt and Bures degrees of polarization

In the figure B.1, we have plotted the Hilbert-Schmidt and Bures degree of po-
larization for some two-dimensional states. From the explanation in the caption, we
see that the two measures order some pairs of states differently. The Hilbert-Schmidt
and Bures distances thus induce two fundamentally different degrees of polarization.


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	Introduction
	Contents
	1. Resumen
	1.1 Sistemas cuánticos discretos
	1.1.1 Sistemas de dos niveles
	1.1.2 Sistemas de tres niveles

	1.2 Fase relativa en interacciones átomo-campo
	1.2.1 Átomos de dos niveles interactuando con campos cuánticos
	1.2.2 Átomos de tres niveles en interacción con campos cuánticos

	1.3Rotaciones pequeñas y hamiltonianos efectivos
	1.4Grado de polarización para campos bimodales

	2. Discrete quantum systems: a cursory look
	2.1 Two-level systems
	2.1.1 General considerations: Bloch sphere
	2.1.2 Systems of qubits
	2.1.3 Phase for a qubit
	2.1.4 Complementarity in the Bloch sphere

	2.2 Three-level systems
	2.2.1 Bloch sphere for a qutrit
	2.2.2 Phases for a qutrit
	2.2.3 Complementarity for a qutrit


	3. Relative phase in atom-field interactions
	3.1 Two-level atoms interacting with a quantum field
	3.1.1 Jaynes-Cummings model
	3.1.2 Relative-phase for the Jaynes-Cummings model
	3.1.3 Dicke model
	3.1.4 Relative-phase for the Dicke model

	3.2 Three-level atoms interacting with quantum fields
	3.2.1 Three-level atom coupled to a two-mode field
	3.2.2 The role of atom-field relative phase
	3.2.3 Relative-phase distribution function


	4. Small rotations and effective Hamiltonians
	4.1 Effective Hamiltonians for nonlinear su(2) dynamics
	4.1.1 Motivation of the method
	4.1.2 Nonlinear small rotations
	4.1.3 Dispersive limit of the Dicke model

	4.2 Effective Hamiltonians for nonlinear su(3) dynamics
	4.2.1 Three-level systems interacting with quantum fields
	4.2.2 The dispersive limit of the  configuration

	4.3 Effective Hamiltonians for nonlinear su(d) dynamics
	4.3.1Multilevel systems interacting with a quantum field
	4.3.2 Three-photon resonance
	4.3.3 Multimode fields and resonance conditions


	5. Relative phase for two-mode fields
	5.1 States with well-defined relative phase
	5.1.1 Differential phase shifts
	5.1.2 Classical visibility
	5.1.3 Quantum visibility

	5.2 SU(2) invariance properties of two-mode fields
	5.3 Quantum degree of polarization as a distance
	5.3.1 Some examples
	5.3.2 Maximally polarized states


	6. Conclusions
	A Positive operator-valued measures
	A.1 Generalized measurements
	A.2 Example: POVMs for angular variables

	B Hilbert-Schmidt and Bures degrees of polarization
	References
	portada1.pdf
	MEMORIA PARA OPTAR AL GRADO DE DOCTOR 
	Madrid, 2007